Second Order Linear DE

For two solutions of $y''+p(t)y'+q(t)y=0$, the Wronskian $W[y_1,y_2](t)=\det \begin{bmatrix}y_1(t) & y_2(t)\\ y_1'(t) & y_2'(t) \end{bmatrix}$ is given by $$\tag{Abel's formula} W[y_1,y_2](t)=C \exp\left(-\int p(t)dt\right) $$

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Compute Wronskian for solutions of equation $t y'' +4 y=0$. Simplify your answer!

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Answer: $W[f,g]=C$
Solution Since there is no $y'$ in the equation, $p(t)=0$
Another one?

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Compute Wronskian for solutions of equation $t y'' +4 y'=0$. Simplify your answer!

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Answer: $W=\frac{C}{t^4}$
Solution $p(t)=4/t$ so $W=C e^{-4\int \frac{1}{t} dt}=C e^{-4 \ln t}=C/t^4$
Another one?

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Compute Wronskian for solutions of equation $t^2 y'' +4 y'+y=0$. Simplify your answer!

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Answer: $W[f,g]=C e^{4/t}$
Solution $p(t)=4/t^2$ so $W=C e^{-\int 4/t^2 dt}=C e^{4/t}$

One more?

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Compute Wronskian for solutions of equation $e^t y'' + y'+e^{-t} y=0$. Simplify your answer!
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Answer: $W=C e^{(e^{-t})}$

Solution $p(t)=e^{-t}$ so $W=C e^{-\int e^{-t} dt}=C e^{(e^{-t})}$
One more?