can anyone explain me the principle to solve:
This equation is first-order linear. There's a procedure called variation of parameters, or the integrating factor method, that you can use to solve this equation. Take a look here for detailed instructions on solving these equations, along with some examples.
Consider a DE of the form
where are all functions of and you hope to solve the DE for . We call this type of DE First-Order Linear.
What we aim to do is to find a function of , which we will call (I use because it's called the Integrating Factor) which will enable us to write the LHS as a single derivative.
Let's say we had multiplied both sides by . That would give
.
Now let's take a closer look at the product rule.
If you wished to find the derivative of , where and are functions of . Using the product rule this would give
.
The LHS of your DE (after multiplying by ) looks suspiciously similar. In fact, if and , then it IS in fact a product rule expansion. Note that if then .
Substituting gives , which is separable, and solving for gives
where .
Any value of will work when we multiply the DE by the integrating factor, so we choose for simplicity.
So you will need to multiply through by your integrating factor, which is of the form , and that will reduce your LHS into a product rule expansion, which will then make the function integrable.
So looking at your example...
.
Notice that your , thus your integrating factor is .
Multiplying through by your integrating factor gives
(check using the product rule...)
.