`xy + y' = 100x` Solve the differential equation

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The problem:` xy+y'=100x` is as first order differential equation that we can evaluate by applying variable separable differential equation:



`N(y) dy=M(x) dx`

Apply direct integration:` intN(y) dy= int M(x) dx` to solve for the

 general solution of a differential equation.

Applying variable separable differential equation, we get:


`y' =100x-xy`


`(y')/(100-y)= x`

Let `y' =(dy)/(dx)` :

`((dy)/(dx))/(100-y)= x`

`(dy)/(100-y)= x dx`

Apply direct integration on both sides:

`int(dy)/(100-y)= int x dx`

For the left side, we consider u-substitution by letting:

`u= 100-y` then `du = -dy` or -`du=dy.`

The integral becomes:


Applying basic integration formula for logarithm:

`int(-du)/(u)= -ln|u|`

Plug-in `u = 100-y` on "`-ln|u|` " , we get:



For the right side, we apply the Power Rule of integration: `int x^n dx = x^(n+1)/(n+1)+C`


 `int x* dx= x^(1+1)/(1+1)+C`


               ` = x^2/2+C`


Combing the results from both sides, we get the general solution of the differential equation as:

`-ln|100-y|= x^2/2+C`


`y =100- e^(-x^2/2-C)`

 `y = 100-Ce^(-x^2/2)

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial