Generals 2001 II 2

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The homogenous equation is:

y^{\prime\prime}+\frac{y}{1+x}=0


The approximate solution for x\rightarrow\infty is:

y_{h}=e^{\pm i\sqrt{1+x}}


The full equation is:

y^{\prime\prime}+\frac{y}{1+x}=x


Guessing the balance for x\rightarrow\infty between y/\left(1+x\right) and x, we get yp = x + x2. A check reveals that this is dominant. Then the general solution is:

y=c_{1}\sin\left(\sqrt{1+x}\right)+c_{2}\cos\left(\sqrt{1+x}\right)+x+x^{2}
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