# Linear differential equation

Homework Statement:
Fing the general solution
Relevant Equations:
Integrating factor = exp{integral[p(x)]}
I tried it but I don't know how to evaluate the integral on the last equation. Help.

#### Attachments

• 20201102_100129.jpg
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## Answers and Replies

Mark44
Mentor
Homework Statement:: Fing the general solution
Relevant Equations:: Integrating factor = exp{integral[p(x)]}

I tried it but I don't know how to evaluate the integral on the last equation. Help.
Can you post a photo of the problem as given in your book?

Delta2
Homework Helper
Gold Member
It seems to me what you did is almost correct (you forgot the integration constant though so that equation should be $$ye^{x^2}=\int x^2e^{x^2}dx+C$$)
However the above integral doesn't have a closed form. Perhaps the original equation is $$\frac{dy}{dx}=x^3-2xy$$? (or even $$\frac{dy}{dx}=x-2xy$$) cause if it is so then the integral will be $$\int x^3 e^{x^2} dx$$ (or $$\int xe^{x^2} dx$$) and will have a closed form.

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Mark44
Mentor
However the above integral doesn't have a closed form.
Which is why I asked to see the original problem, in case it's different from what is being worked on here.

• Delta2
It seems to me what you did is almost correct (you forgot the integration constant though so that equation should be $$ye^{x^2}=\int x^2e^{x^2}dx+C$$)
However the above integral doesn't have a closed form. Perhaps the original equation is $$\frac{dy}{dx}=x^3-2xy$$? (or even $$\frac{dy}{dx}=x-2xy$$) cause if it is so then the integral will be $$\int x^3 e^{x^2} dx$$ (or $$\int xe^{x^2} dx$$) and will have a closed form.
I've attached the original pocture sir. It's really x^2. So does that mean the differential equation can't be solved? And by means "closed form" what does it mean sir. I want to know more. Tnx in advance.

#### Attachments

Delta2
Homework Helper
Gold Member
By closed form we mean a form that is finite and contains only the elementary well known functions (##x^n,\sin x, e^x,\ln x,a^x## e.t.c) for example $$y=\frac{\tan(x^2e^{x^2}+1)+\ln({x^3+1})}{2^x}$$ is a closed form (no matter how complex it might be).

In this case we cannot express the integral as a closed form so all we can do is write the integral in the solution just as it is, so the solution would be $$y=e^{-x^2}\left ( \int x^2e^{x^2} dx+C\right )$$ which is a solution not in closed form. So a solution exists , but simply we cannot write the solution in closed form.

I suspect a typo in the statement of the problem, it should be ##x## or ##x^3##, cant explain it otherwise.

Last edited:
Mark44
Mentor
I suspect a typo in the statement of the problem
I do, too, unless the intent was to give the solution in terms of the integral shown in Delta2's post.