## Related questions with answers

Question

The given differential equation has a fundamental set of solutions whose Wronskian $W(t)$ is such that $W(0)=1$. What is $W(4)$ ?

$y^{(4)}-y^{\prime \prime}+y=0$

Solution

VerifiedStep 1

1 of 2From $\textsf{Abel's Theorem}$ (see equation $(7)$) we have that since $W(0)=1$ we know that the value of the Wronskian of a set of solutions of the differential equation

$y^{(4)} - y^{\prime\prime} +y =0$

will be:

$\pmb{W(t)} = W(0) e^{ -\int_0^t 0 ds } = 1 e^{0 }= \pmb{1 }$

We had the constant function $0$ in the integral because $y^{\prime\prime\prime}$ does not appear in the equation !

We conclude that $\boxed{ \pmb{ W(4) = 1}}$

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