Differential equations with variables separable

Differential equations with variables separable

Edited By Komal Miglani | Updated on Jul 02, 2025 06:36 PM IST

Before we learn the concept of the variable separable form of the differential equation, let's first understand what a differential equation is. A differential equation is an equation involving one or more terms and the derivatives of one dependent variable with respect to the other independent variable. Differential equations with separable variables are types of special classes of first-order differential equations that can be solved by separating the variables and integrating both sides independently. This method is very straightforward and widely used because it simplifies the process of finding the solution. Due to this, these types of differential equations are easily solved as compared to other types of differential equations. It also helps in solving real-life problems like calculating the population growth of the country and finding radioactive decay.

This Story also Contains
  1. What is a Differential Equation?
  2. Separable Differential Equation
  3. Variable Separable Differential Equation Definition
  4. Solved Examples Based On Variable Separable Differential Equations
  5. Summary
Differential equations with variables separable
Differential equations with variables separable

In this article, we will cover the concept of differential equations with variable separable methods. This concept falls under the broader category of differential equations, which is a crucial chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of nineteen questions have been asked on this concept, including one in 2015, one in 2017, one in 2019, five in 2020, six in 2021, four in 2022, and one in 2023.

What is a Differential Equation?

A differential equation is an equation involving one or more terms and the derivatives of one dependent variable with respect to the other independent variable.

Differential equation: dy/dx = f(x)

Where “x” is an independent variable and “y” is a dependent variable

Example of differential equation: $x \frac{d y}{d x}+2 y=0$
The above-written equation involves variables as well as the derivative of the dependent variable $\mathrm{y}$ with respect to the independent variable $\mathrm{x}$. Therefore, it is a differential equation.
The following relations are some of the examples of differential equations:
(i) $\frac{d y}{d x}=\sin 2 x+\cos x$
(ii) $\mathrm{k} \frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}=\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^2\right]^{3 / 2}$

Separable Differential Equation

Differential equations where the variables can be separated from each other are called separable differential equations. The general form of a separable differential equation is dy/dx = f(x)g(y).

These equations can be easily solved by separating the variables and integrating them individually.

Variable Separable Differential Equation Definition

The differential of the form $\frac{d y}{d x}=f(x) g(y)$ where $f(x)$ is a function of $x$ and $g(y)$ is a function of $y$, are said to be variable separable form.

Steps to Solve differential equations using variable separable methods-

  1. Check for any values of y that make g(y)=0. These correspond to constant solutions.
  2. Rewrite the differential equation in the form $\frac{d y}{g(y)}=f(x) d x$
  3. Integrate both sides of the equation.
  4. Solve the resulting equation for y if possible.
  5. If an initial condition exists, substitute the appropriate values for x and y into the equation and solve for the constant.

Rewrite the equation as
$
\frac{d y}{g(y)}=f(x) d x \quad[\text { where } g(y) \neq 0]
$

This process is separating the variables. Now, integrating both sides, we get
$
\int \frac{d y}{g(y)}=\int f(x) d x+c
$

By this, we get the solution of the differential equation

Let's see some illustration for a better understanding
(i) Solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=\left(\mathrm{e}^{\mathrm{x}}+1\right)\left(\mathrm{y}^2+1\right)$

Rewrite the differential equation as
$
\frac{\mathrm{dy}}{1+\mathrm{y}^2}=\left(\mathrm{e}^{\mathrm{x}}+1\right) \mathrm{dx}
$

Integrating both sides, we get
$
\begin{aligned}
& \int \frac{\mathrm{dy}}{1+\mathrm{y}^2}=\int\left(\mathrm{e}^{\mathrm{x}}+1\right) \mathrm{dx} \\
& \Rightarrow \tan ^{-1} y=e^x+x+c \\
& \Rightarrow y=\tan \left(e^x+x+c\right)
\end{aligned}
$

(ii) A differential equation of the form $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{f}(\mathrm{ax}+\mathrm{by}+\mathrm{c})$ where $\mathrm{a}, \mathrm{b}$, and $\mathrm{c}$ are constants, can be converted into an equation with variables separable by the substitution $\mathrm{v}=\mathrm{ax}+\mathrm{by}+\mathrm{c}$.
$
\begin{aligned}
& \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{f}(\mathrm{ax}+\mathrm{by}+\mathrm{c}) \\
& \mathrm{v}=\mathrm{ax}+\mathrm{by}+\mathrm{c} \\
& \therefore \frac{d v}{d x}=a+b \frac{d y}{d x} \text { or, } \frac{d y}{d x}=\frac{\frac{d v}{d x}-a}{b} \\
& \Rightarrow \frac{\frac{\mathrm{d} v}{\mathrm{dx}}-\mathrm{a}}{\mathrm{b}}=\mathrm{f}(\mathrm{v}) \Rightarrow \frac{\mathrm{dv}}{\mathrm{dx}}=\mathrm{bf}(\mathrm{v})+\mathrm{a} \\
& \Rightarrow \frac{\mathrm{dv}}{\mathrm{bf}(\mathrm{v})+\mathrm{a}}=\mathrm{dx}
\end{aligned}
$

In the differential equation (ii), the variables $\mathrm{x}$ and $\mathrm{v}$ are separated.
Integrating (ii), we get
$
\begin{aligned}
& \Rightarrow \quad \int \frac{d v}{b f(v)+a}=\int d x+C \\
& \Rightarrow \quad \int \frac{d v}{b f(v)+a}=x+C \text {, where } v=a x+b y+c
\end{aligned}
$

This represents the general solution of the differential equation (i).

Recommended Video Based on Variable Separable Differential Equations


Solved Examples Based On Variable Separable Differential Equations

Example 1: Which of the following can be solved using the variable separation method?

(1) $\frac{d y}{d x}=1+x y$
(2) $\frac{d y}{d x}=x^2 y$
(3) $\frac{d y}{d x}=\sin (x+y)$
(4) $\frac{d y}{d x}=x^y$

Solution:
In options (1),(3), and (4), $\mathrm{x}$ and $\mathrm{y}$ cannot be completely separated to make quantity involving $\mathrm{x}$ with $\mathrm{dx}$ and quantity involving $y$ with dy. But it can be done in option (2) as:
$
\frac{d y}{d x}=x^2 y \Rightarrow \frac{d y}{y}=x^2 d x
$

Hence, the answer is the option (2).

Example 2: Which of the following doesn't represent variable separated form?
(1) $\sin x d x+\cos y d y=0$
(2) $\sin (x+y) d x+\cos y d y=0$
$\left(1+x^2\right) d x+\frac{1}{y} d y=0$
(4) $\sec ^2 x d x+\tan y d y=0$

Solution:
The differential equations in options (1), (3) and (4) are of variable separable form.
But in option (2), we have:
$
\sin (x+y) d x+\cos y d y=0
$

In this, variables cannot be separated as $\sin (\mathrm{x}+\mathrm{y})$ is a composite function.
Hence, the answer is the option (2).

Example 3: Given that the slope of the tangent to a curve $y=y(x)$ at any point $(x, y)$ is $\frac{2 y}{x^2}$. If the curve passes through the centre of the circle $x^2+y^2-2 x-2 y=0$, then its equation is:

Solution:
Given the slope of the tangent $=\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{2 y}{x^2}$.
$
\frac{\mathrm{d} y}{\mathrm{y}}=\frac{2 d x}{\mathrm{x}^2}
$

Integrate both sides.
$
\ln (y)=-\frac{2}{x}+C
$

It passes through the center of the circle $x^2+y^2-2 x-2 y=0$ i.e $(1,1)$
$
0=-2+C \Rightarrow C=2
$

Eq. of the curve is
$
\begin{aligned}
& x \cdot \ln |y|=-2+2 x \\
& x \cdot \ln |y|=2(x-1)
\end{aligned}
$

Hence, the answer is $x \log _e|y|=2(x-1)$

Example 4: The solution of the differential equation $x\left(y^2+y\right) d x+(x+1) d y=0$ is:
Solution:
$
x y(y+1) d x+(x+1) d y=0
$

Dividing throughout by $y(y+1)(x+1)$, we get:
$
\begin{aligned}
& \frac{x d x}{(x+1)}+\frac{d y}{y(y+1)}=0 \\
& \Rightarrow \int \frac{x d x}{(x+1)}+\int \frac{d y}{y(y+1)}=c \\
& \Rightarrow \int \frac{(x+1)-1}{(x+1)} d x+\int \frac{(1+y)-y}{y(y+1)} d y=c \\
& \Rightarrow \int\left(1-\frac{1}{(x+1)}\right) d x+\int\left(\frac{1}{y}-\frac{1}{y+1}\right) d y=c \\
& \Rightarrow x-\ln |x+1|+\ln |y|-\ln |y+1|=c \\
& \Rightarrow \ln \left|\frac{y}{(x+1)(y+1)}\right|=c-x \\
& \Rightarrow\left|\frac{y}{(x+1)(y+1)}\right|=e^{c-x} \\
& \Rightarrow\left|\frac{y}{(x+1)(y+1)}\right|=c \cdot e^{-x} \\
&
\end{aligned}
$

Hence, the answer is $\left|\frac{y}{(x+1)(y+1)}\right|=c \cdot e^{-x}$

Example 5: Let $y=f(x)$ be the solution of the differential equation $y(x+1) d x-x^2 d y=0, y(1)=e$. Then $x \rightarrow 0^{+} f(x)$ is equal to

Solution:
$
\begin{aligned}
& y(x+1) d x-x^2 d y=0, \quad y(1)=e \\
& \Rightarrow \frac{d y}{d x}=\frac{y(x+1)}{x^2} \\
& \Rightarrow \int \frac{d y}{y}=\int \frac{(x+1) d x}{x^2} \\
& \ell \text { ny }=\ell \mathrm{nx}-\frac{1}{x}+c \\
& \because y(1)=e \\
& \therefore 1=0-1+C \Rightarrow C=2
\end{aligned}
$

Now, $\ell \mathrm{ny}=\ell \mathrm{nx}-\frac{1}{\mathrm{x}}+2$
$
\begin{aligned}
& \Rightarrow \ell \ln \left(\frac{y}{x}\right)=2-\frac{1}{x} \\
& \Rightarrow \frac{y}{x}=e^{2-\frac{1}{x}} \\
& \Rightarrow y=x, e^{2-\frac{1}{x}}
\end{aligned}
$

So, $\lim _{x \rightarrow 0^{+}} y=\lim _{x \rightarrow 0^{+}} x e^{2-\frac{1}{x}}=0$
Hence, the answer is 0.

Summary

Variable Separable differential equations are a useful and accessible class of differential equations that can be equated in many real-life phenomena, such as population growth and radioactive decay. By separating the variables and integrating both sides independently, we can find solutions that describe how these systems evolve. This method is a fundamental tool in the analysis of dynamic systems in various fields, including mathematics, biology, chemistry, and physics.

Frequently Asked Questions (FAQs)

1. What is a differential equation?

It describes the rate of change in quantity and is used in science, engineering, business, etc. It can model many phenomena in different fields.

2. Define variable separable differential equation.

 A variable separable differential equation is a first-order differential equation that can be written in the form dy/dx=k(y)h(x). The variables k(y) and h(y) can be separated on opposite sides of the equation, allowing for individual integration.

3. How to Identify Separable Differential Equations?

 To identify the variable separable differential equations easily by separating the independent variable or term and dependent variable on either side of the equal sign. If we are able to separate those variables then the equation is a separable differential equation.

4. How to solve a Separable Differential Equation?

Separable differential equations are easily solved by arranging all the dependent variables on one side of the equal sign and the independent variable on the opposite side of the equal sign to separate the different terms and then integrating both sides separately to get the final answer.

5. What are Non-Separable Differential Equations?

All the differential equations where we can not separate the independent variable (x) and the dependent variable (y) on either side of the equal sign are called non-separable differential equations.

6. Why are separable differential equations important?
Separable differential equations are important because they represent many real-world phenomena and are often the simplest type of differential equations to solve. They serve as a foundation for understanding more complex differential equations and provide a starting point for modeling various physical, biological, and economic systems.
7. How does the process of separation of variables work?
The process of separation of variables involves rearranging the differential equation so that all terms containing x and dx are on one side, and all terms containing y and dy are on the other side. This allows each side to be integrated separately, simplifying the solution process.
8. What is the relationship between separable differential equations and implicit functions?
The solution to a separable differential equation is often expressed as an implicit function, where y is not explicitly solved for in terms of x. This implicit form represents the relationship between x and y that satisfies the original differential equation.
9. How do you solve a separable differential equation when one side is a fraction?
When one side of a separable differential equation is a fraction, you can multiply both sides by the denominator to simplify the equation before separating variables. This step helps in preparing the equation for integration.
10. What are some real-world applications of separable differential equations?
Separable differential equations have numerous applications, including:
11. How do you solve a separable differential equation when it involves natural logarithms?
When a separable differential equation involves natural logarithms:
12. What are differential equations with variables separable?
Differential equations with variables separable are a type of first-order differential equation where the variables can be separated so that all terms involving one variable are on one side of the equation, and all terms involving the other variable are on the other side. This allows for easier integration and solving of the equation.
13. How do you identify a separable differential equation?
A separable differential equation can be identified when it can be written in the form dy/dx = f(x)g(y), where f(x) is a function of x only and g(y) is a function of y only. If the equation can be rearranged into this form, it is separable.
14. What is the general method for solving separable differential equations?
The general method for solving separable differential equations involves these steps:
15. Can all differential equations be separated?
No, not all differential equations can be separated. Only certain types of first-order differential equations are separable. Many differential equations require other solution methods, such as integrating factors, substitution, or numerical methods.
16. What is the difference between separable and non-separable differential equations?
Separable differential equations can be written in the form dy/dx = f(x)g(y), where variables can be separated onto different sides of the equation. Non-separable equations cannot be written in this form and require different solution techniques.
17. What is an autonomous differential equation and how does it relate to separable equations?
An autonomous differential equation is one where the independent variable (usually time) does not appear explicitly. Many autonomous equations are separable, as they can often be written in the form dy/dx = f(y). This connection makes separable equations particularly useful in studying time-independent systems.
18. What is the relationship between separable differential equations and exact differential equations?
While not all separable equations are exact, and not all exact equations are separable, there is an overlap:
19. How do initial conditions affect the solution of a separable differential equation?
Initial conditions provide specific values for the variables at a particular point, allowing the determination of the constant of integration in the general solution. This transforms the general solution into a particular solution that satisfies the given initial conditions.
20. What is the difference between the general and particular solution of a separable differential equation?
The general solution of a separable differential equation includes a constant of integration and represents all possible solutions. A particular solution is obtained by using specific initial conditions to determine the value of the constant, resulting in a single, specific solution that satisfies those conditions.
21. How do you interpret the direction field of a separable differential equation?
The direction field of a separable differential equation provides a visual representation of the equation's solutions:
22. What is the significance of the constant of integration in separable differential equations?
The constant of integration represents the family of solutions to the differential equation. It allows for the general solution to be adapted to specific initial conditions, making it possible to find particular solutions that satisfy given constraints.
23. Can separable differential equations have multiple solutions?
Yes, separable differential equations can have multiple solutions. The general solution often includes a constant of integration, which can lead to an infinite family of solutions. Specific initial conditions are usually needed to determine a unique solution.
24. How do you handle a separable differential equation when g(y) = 0?
When g(y) = 0 in a separable differential equation dy/dx = f(x)g(y), it indicates a constant solution where y does not change with respect to x. This special case should be considered separately from the general solution obtained through integration.
25. What role does the chain rule play in solving separable differential equations?
The chain rule is implicitly used when separating variables. It allows us to write dy/g(y) on one side of the equation, which can then be integrated with respect to y. Understanding the chain rule is crucial for correctly manipulating and solving these equations.
26. How do you graph solutions to separable differential equations?
Graphing solutions to separable differential equations often involves:
27. How do you handle separable differential equations with trigonometric functions?
When dealing with separable differential equations containing trigonometric functions:
28. What is the significance of equilibrium solutions in separable differential equations?
Equilibrium solutions are constant solutions where the rate of change is zero. In separable equations of the form dy/dx = f(x)g(y), equilibrium solutions occur when g(y) = 0. These solutions are important as they represent steady states of the system and can provide insights into long-term behavior.
29. How do you solve a separable differential equation when it involves absolute values?
When a separable differential equation involves absolute values:
30. How do you handle separable differential equations with polynomial functions?
For separable differential equations with polynomial functions:
31. What is the importance of domain restrictions in solutions to separable differential equations?
Domain restrictions in solutions to separable differential equations are crucial because:
32. How do you solve a system of separable differential equations?
To solve a system of separable differential equations:
33. How do you determine the long-term behavior of solutions to separable differential equations?
To determine the long-term behavior of solutions:
34. How do you handle separable differential equations with piecewise-defined functions?
For separable differential equations with piecewise-defined functions:
35. How do you analyze the stability of equilibrium solutions in separable differential equations?
To analyze the stability of equilibrium solutions:
36. What is the importance of dimensional analysis in solving separable differential equations?
Dimensional analysis is important in solving separable differential equations because:
37. How do you handle separable differential equations that involve inverse trigonometric functions?
When dealing with separable differential equations involving inverse trigonometric functions:
38. What is the role of symmetry in solving certain separable differential equations?
Symmetry can play an important role in solving separable differential equations:
39. How do you determine if a separable differential equation has a unique solution?
A separable differential equation has a unique solution if:
40. What are singular solutions in the context of separable differential equations?
Singular solutions are solutions to a differential equation that cannot be obtained from the general solution by specifying a value for the constant of integration. In separable equations, these often arise when g(y) = 0, leading to constant solutions that may not be part of the general solution family.
41. What are homogeneous differential equations and how do they relate to separable equations?
Homogeneous differential equations are those where the right-hand side is a homogeneous function of x and y. While not all homogeneous equations are separable, many can be transformed into separable equations by substituting y = vx, where v is a new variable. This connection provides a bridge between these two types of equations.
42. How do separable differential equations relate to the concept of conservation laws in physics?
Many conservation laws in physics can be expressed as separable differential equations:
43. What are the limitations of the separation of variables method?
The separation of variables method has several limitations:
44. What is the role of implicit differentiation in understanding separable differential equations?
Implicit differentiation is important in understanding separable differential equations because:
45. What is the significance of the separation constant in certain separable differential equations?
The separation constant, often denoted as λ, appears in some separable differential equations, particularly in partial differential equations that can be solved by separation of variables. Its significance includes:
46. What is the relationship between separable differential equations and variables in calculus?
The relationship between separable differential equations and variables in calculus is fundamental:
47. What are some common mistakes students make when solving separable differential equations?
Common mistakes include:
48. How do separable differential equations relate to the concept of exponential growth and decay?
Separable differential equations are closely related to exponential growth and decay:

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