Solve the differential equation: .

Or

Solve the differential equation: ;y(0)=0.

Given that

This differential equation is of the form y’ + P(x)y = Q(x).

To solve such equation, we multiply the entire equation with integration factor .

Here

Then integration factor will be .

Let 1+x²=t

⇒ 2x dx=dt.

Multiplying (1) with , we get

⇒

Integrating both sides

=tan^-1 x + x + c


y = (1+x²)(tan^-1 x + x + c)

OR

Given that

Integrating both sides

Let 2-e^y=t

⇒ e^ydy=-dt

or

Hence ,
where

Or

Since bases of both logs are same, therefore the argument must be equal too

Simplifying this equation, we get

Since when x=0, y=0, therefore

or k=1

Hence,