How to Solve Linear Differential Equations

By Prashant Kumar|Updated : August 19th, 2018

Form of linear differential equation is

Here P(x) and Q(x) are the function of x.

To solve this type of differential equation we use the integration factor. We don’t need the derivation on how we got the integration factor. Integration factor is given by

Now both sides of the equation with IF and simplify:

So the Final Form that we have to solve is

Now we have to use simple integration concept and find the answer.



Solution: Simplify this equation to get

We can see this is linear differential equation. So we find the integration factor first.

So the solution of this equation is given by

Here C is the integral constant.


We see how to solve a linear differential equation. Now here we see a different type of equation and that can be reducible to Linear differential equation form.


To solve this, we use substitution method and convert this into linear form.

So the equation is

And this is Linear differential equation. Now we can solve this using linear differential method. Here we find the solution in u and x term. Then put back value of u = f(y).


Example: Solve

Solution: -  We can write this equation as

Now here we put


New equation is

And this is a linear differential equation. Find the IF of this equation

So the solution of the equation is given by

Put back the value of v


The basic form of exact differential equation is

M(x, y) dx + N(x, y) dy = 0

Here M and N are function of x and y such that

If this is exact differential equation then there exist a function f (x, y) such that

And once we find that function then the general solution is

                                    f(x, y) = c

So to find this we use know terms.


Finally, to find the solution we ‘merge’ both the functions obtained above. To do that, write down each term exactly once, even if a particular term appears in both results. And that merged form is called f(x, y) and the final solution is

f(x, y) = c



Solution: Here we see that

M = 2xy - 9x2

N = 2y + x+ 1

And if we find

Here we can see that

My = Nx

So this is an exact differential equation.

Now we find the solution.

So for final answer we merge both solution we find. Here we can see that x2y is the common term in both solution so we write that only once.

f(x,y) = x2y - 3x+ y+ y

So the general solution is

This is the final solution of the given exact differential equation. Here c is the integral constant.


Solved Examples of Differential Equations

  1. Solve the differential equation 


1st we write this in separate from.

Now integrate both side.

So this is the final solution of differential equation.



  1. Find the curve for which the intercept cut off by any tangent on y-axis is proportional to the square of the ordinate of the point of tangency.


Let p(x, y) be any point on the curve

Length of intercept on y axis = y - x (dy/dx)

And by the question the length of intercept on y-axis is proportional to y2



Here k is the constant of proportionality.

So the simple form of the equation is

Put 1/y = v


So the differential equation is

And this is a linear differential equation

So the solution is

Now put back v=1/y


So this is the final answer. Here C is the integral constant and k is the proportional constant.


  1. Find the orthogonal trajectories of family of curves x2 + y2 = cx

Answer: So if 2 functions are orthogonal then the product of slope at any point curve is -1. Because they make 90-degree angle with each other. Now we find the slope (dy/dx) at any point (x,y) for the given curve and find the slope for the orthogonal curve and solve that.

Now if the slope of other curve is M then

So we take the slope of other curve as dy/dx, so the differential equation is

And this is homogeneous equation so put

y =vx

Here k = 1/C and it is a constant.

So the orthogonal curve to the given curve is  x+ y= ky


  1. Solve 

Answer: We can write this as

So here we can take (y+1)/x as a another variable and then solve it.


So the equation is

Click on the links below to read more about Differential Equations:

How to Solve Homogeneous Differential Equation

How to Solve Differential Equations by Variable Separable Method

Order, Degree & Solutions of Differential Equations


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
Raees Gaurav Singh
Amazing sir 👏👏👏🙏🙏🙏🙇
Shrey Vimal

Shrey VimalMar 16, 2020

This way of uploading articles regarding separate topic is exactly what i was searching in an app for got something imperesive going on here...keep it up guys :)  
...Read More

Follow us for latest updates