DE x*y' + x^2*y = x^2 y(0) = 2

[Riazy]

x*y' + x^2*y = x^2 y(0) = 2

Hi guys this is the problem


My attempt:

1. x*y' + x^2*y = x^2
Looking at this problem, I can first see that it is of the form: dy/dx + a(x)*y=b(x)

The aim here is to get the Integrating factor of a(x) and then multiply it by the DE.

I cannot do that until I have done this

y'+x*y = x, that is removing the constant of y'

2. Now to the Integrating factor [shortned I.F] an I.F is e^$a(x)dx
which in my new case would be a(x) = x, thus: e^$xdx = e^x^2/2

3. Multiplying the IF by the whole DE gives us.

(e^x^2/2) * y' + (e^x^2/2) * x*y = (e^x^2/2) * x

The thing is that I can't figure out how to solve this

can anyone sovle this for me please, Thanks!

 

[tkhunny]

You already solved it. All that is left is to see it.

What is the point of the Integrating Factor? It builds for you the pieces of a derivative by product rule. Don't try to figure out what the product is. Just write it down. y*(integrating factor) - every time. You're almost done.

 

[Deleted member 4993]

x*y' + x^2*y = x^2 y(0) = 2

Hi guys this is the problem


My attempt:

1. x*y' + x^2*y = x^2
Looking at this problem, I can first see that it is of the form: dy/dx + a(x)*y=b(x)

The aim here is to get the Integrating factor of a(x) and then multiply it by the DE.

I cannot do that until I have done this

y'+x*y = x, that is removing the constant of y'

2. Now to the Integrating factor [shortned I.F] an I.F is e^$a(x)dx
which in my new case would be a(x) = x, thus: e^$xdx = e^x^2/2

3. Multiplying the IF by the whole DE gives us.

(e^x^2/2) * y' + (e^x^2/2) * x*y = (e^x^2/2) * x
The thing is that I can't figure out how to solve this

can anyone sovle this for me please, Thanks!

After you integrate both sides - the left hand side becomes y*(integrating factor). Right hand side is the part new for different problems.

So in your case it becomes

e^(x[sup:tbgojjkl]2[/sup:tbgojjkl]/2) * y = e^(x[sup:tbgojjkl]2[/sup:tbgojjkl]/2) + C

.