I have a basic differential equation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{dy}{dx} = x + y, y(0) = 1[/tex]

Now, when I try to solve this by making it exact

[tex]\mu \frac{dy}{dx} + \mu y = \mu x[/tex]

I get [tex]\mu = e^{-x}[/tex] and solution [tex]-x-1[/tex]. This doesn't satisfy the initial condition. But when I try to solve it as a non-homogenous equation as:

[tex]\frac{dy}{dx} + y= x[/tex]

I get

[tex]y_p = 2e^x, y_c = -x-1[/tex]

so

[tex]y = 2e^x-x-1[/tex]

Which seems to be a correct & full solution. What was I missing in the first try?

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# Solution conflict!

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