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?
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.
If You write the DE as...
$\displaystyle y^{'} = -2xy +1$ (1)
... it is easy to see that it's a linear DE on the form...
$\displaystyle y^{'} = a(x)\cdot y +b(x)$ (2)
... where $\displaystyle a(x)= -2x$ and $\displaystyle b(x)=1$ and its solution can be found is 'standard fashion' ...
$\displaystyle y= e^{\int a(x)\cdot dx} \{\int b(x)\cdot e^{-\int a(x)\cdot dx }\cdot dx +c\} = e^{-x^{2}}\cdot (\int e^{x^{2}}\cdot dx + c)$ (3)
The integral in (3) is not elementary but it can be expressed as a convergent series...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$