# Verify Solutions of DEs Exercises

## Exercises

Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function.

1. {$$ \large y=ce^{2x}\;; \quad y'=2y$$}

2. {$$ \large y={x^2\over3}+{c\over x}\;; \quad xy'+y=x^2 $$}

3. {$$ \large y={1\over2}+ce^{-x^2}\;; \quad y'+2xy=x $$}

4. {$$ \large \begin{multline}y=\left(1+ce^{-x^2/2}\right) \left(1-ce^{-x^2/2}\right)^{-1}; \cr \shoveleft 2y'+x(y^2-1)=0 \end{multline} $$}

5. {$$ \large \begin{multline}y={\tan\left( {x^3\over3}+c\right)}; \cr \shoveleft y'=x^2(1+y^2) \end{multline} $$}

6. {$$ \large \begin{multline}y=(c_1+c_2x)e^x+\sin x+x^2\;; \cr \shoveleft y''-2y'+y=-2 \cos x+x^2-4x+2 \end{multline}$$}

7. {$$ \large \begin{multline}y=c_1e^x+c_2x+ {2\over x}\;; \cr \shoveleft (1-x)y''+xy'- y=4(1-x-x^2)x^{-3} \end{multline}$$}

8. {$$ \large \begin{multline}y=x^{-1/2}\;(c_1\sin x+c_2 \cos x)+4x+8\;; \cr \shoveleft x^2y''+xy'+ {\left(x^2-{1\over4}\right)}y=4x^3+8x^2+3x-2 \end{multline}$$}

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