How do you solve xy'+2y=4x^2 given y(1)=0?

Kwenze0l 2022-09-28 Answered
How do you solve x y + 2 y = 4 x 2 given y(1)=0?
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Answers (1)

Corbin Hanson
Answered 2022-09-29 Author has 10 answers
We have:

x y + 2 y = 4 x 2

Which we can write as:

y + 2 x y = 4 x         ... ... [ 1 ]

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 )

Then the integrating factor is given by;

I F = e P ( x ) d x
= exp (   2 x   d x )
= exp ( 2 ln x )
= e ln x 2
= x 2

And if we multiply the DE [1] by this Integrating Factor, IF, we will have a perfect product differential;

y + 2 x y = 4 x

x 2 y + 2 x y = 4 x 3

d d x ( x 2 y ) = 4 x 3

Which we can directly integrate to get:

  d d x ( x 2 y )   d x =   4 x 3   d x
x 2 y = x 4 + C

Applying the initial condition, y(1)=0, we get:

1 0 = 1 + C C = - 1

Thus the solution is:

  x 2 y = x 4 - 1
y = x 2 - 1 x 2
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