How to Solve Differential Equations by Variable Separable Method

By Prashant Kumar|Updated : August 17th, 2018

If the coefficient of dx is only function of x and coefficient of dy is only a function of y in the given differential equation, then we can separate both dx and dy terms and integrate both separately. And the general form for this method is

f(x) dx+ g(y) dy = 0

And we can find the solution by integrating both sides

Example: Solve 



So this is our final answer. Here we take the as constant as ln(c) just for easier simplification because c and ln(c ) both are arbitrary constant.



Let’s assume a differential equation:

Here f(ax+by+c) is some function of “ax+by+c”

To solve this type of question, put

Now the differential equation is

Now, x and z are separated, so we can integrate them

After solving this we put back z = ax+by+c. And this gives us the general solution.

Here we write a general substitution for ‘ax+by+c’, In some question c=0.


There is another type of problem that can be solved by this method.

Here we can put z=ax+by, where a and b can be found by seeing the question.


Example: Solve


To solve this, we put z = x+y

So by differentiation,

So the New equation is

Integrate both sides,

Now put back value of z = x+y


(x+y)-ln(x+y) = 2x+c

y = x + ln (x + y) + c



If we can write a differential equation in form of

f (f1(x, y)) d(f1(x, y)) + g(f2(x, y)) d(f2(x, y)) + ….. = 0

The we can integrate each term separately. For this we need to remember some direct results.

  1. d(x + y) = dx + dy
  2. d(xy) = x dy + y dx

By using these we can solve some differential equation very quickly. But there is no way to recognize these forms in the questions. We just have to remember these forms and by practice we can find these patterns.


Solution: - First we simplify this

Now if we see this simplified equation then we can see that here we can use some general from of separation.

So the differential equation is

We get,

Here we can see that if we take (x2+y2) as a single variable then on the left side we have a simple integral and similarly, in the right side if we take (x/y) as a single variable.

Let’s put


Now integrate it

Now, substitute the value of v and z,

So the final solution of the differential equation is

So this solution is very short. But if we solve it another way it will be large or sometime we can’t find right form to use.

So these forms are very important and used for tricky and good questions.

In these type of question, we can’t see pattern directly, sometime we have to modify it or if we have solved a lot of example of these type then we can spot these forms.

Click to read about order & degree of differential equations:

Order & Degree of Differential Equations

How to Solve Homogeneous Differential Equation

How to Solve Linear Differential Equations

Stay tuned for more.


Download Gradeup, the best IIT JEE Preparation App
Attempt subject wise questions in our new practice section

All the best!!

Team Gradeup


Posted by:

Prashant KumarPrashant KumarMember since Jun 2018
Life is a journey, not a destination. Enjoy it!
Share this article   |


write a comment
Load Previous Comments
Raees Gaurav Singh
Please upload work power and energy chapter tips
Avinash Kumar
Added few more formulas at the end. Check it out
Anuradha Pandey
Thank you sir
Vivek Arya

Vivek AryaSep 7, 2017

Method to solve cubic equation
rithik singh

rithik singhAug 18, 2018

Sir this help me a lot
Muskan Kumari
It really helped alot thanks

Follow us for latest updates