How can i solve this differencial equation?: y &#x2032; </msup> + x <mrow

Angelique Horne

Angelique Horne

Answered question

2022-05-13

How can i solve this differencial equation?: y + x 2 y = x 2

Answer & Explanation

Giancarlo Shah

Giancarlo Shah

Beginner2022-05-14Added 12 answers

We can use an integrating factor when we have a First Order Linear non-homogeneous Ordinary Differential Equation of the form;
d y d x + P ( x ) y = Q ( x )
We have:
y = x 2 y = x 2 ...[1]
This is a First Order Ordinary Differential Equation in Standard Form. So we compute and integrating factor, I, using;
I = e P ( x ) d x
= exp ( x 2 d x )
= exp ( 1 3 x 3 )
=e^{\fra = e 1 3 x 3
And if we multiply the DE [1] by this Integrating Factor, I, we will have a perfect product differential;
y e 1 3 x 3 + x 2 e 1 3 x 3 y = x 2 e 1 3 x 3
d d x ( y e 1 3 x 3 ) d x + C
Leading to the explicit General Solution:
y = e 1 3 x 3 { e 1 3 x 3 + C }
= 1 + C e 1 3 x 3
vilitatelp014

vilitatelp014

Beginner2022-05-15Added 6 answers

y + x 2 y = x 2
This is a separable ODE
d y d x = x 2 x 2 y = x 2 ( 1 y )
1 1 y d y = x 2 d x
ln | 1 y | = 1 3 x 3 + c
1 y = B e 1 3 x 3 where B = e c
y = 1 + A e 1 3 x 3 where A=-B

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?