Simple differential equation

**daskywalker** · January 16th 2010, 06:25 AM

Find a family of solutions to the DE dy/dx+ 2xy=1

I have some trouble separating the variables...anyone can give me a general method on how to do that?

**Calculus26** · January 16th 2010, 06:41 AM

The trouble is this is not a separable equation.

You have to use an integrating factor .

Do you know this method ?

**daskywalker** · January 16th 2010, 07:13 AM

oh sorry let me rephrase the question:

Verify that the indicated family of functions is a solution to the DE dy/dx+2xy=1:

y= (e^-x^2)*integral from x to 0 (e^(t^2)dt)+ce^(-x^2)

where c is an arbitrary constant.
I suppose the e^t^2 is the integrating factor??
I tried to substitute the y equation into the DE but it does not work.

**Calculus26** · January 16th 2010, 07:40 AM

See the attachment

By the way The integrating factor would be e^(x^2) but you'll get to that later I assume

**chisigma** · January 16th 2010, 08:22 AM

If You write the DE as...

$y^{'} = -2xy +1$ (1)

... it is easy to see that it's a linear DE on the form...

$y^{'} = a(x)\cdot y +b(x)$ (2)

... where $a(x)= -2x$ and $b(x)=1$ and its solution can be found is 'standard fashion' ...

$y= e^{\int a(x)\cdot dx} \{\int b(x)\cdot e^{-\int a(x)\cdot dx }\cdot dx +c\} = e^{-x^{2}}\cdot (\int e^{x^{2}}\cdot dx + c)$ (3)

The integral in (3) is not elementary but it can be expressed as a convergent series...

Kind regards

$\chi$ $\sigma$

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