(a) Solve $\frac{d^{3} y}{d x^{3}}+2 \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-2 y=0$ given $y=0, \frac{d y}{d x}=0, \frac{d^{2} y}{d x^{2}}=6$ when $x=0$ using Laplace transform.

Explanation

This problem requires solving a third-order linear homogeneous differential equation using Laplace transform. The first step is to take the Laplace transform of the differential equation. Then, we substitute the initial conditions and solve for $Y(s)$. Finally, we take the inverse Laplace transform to find the solution $y(x)$.

(a) Solve $\frac{d^{3} y}{d x^{3}}+2 \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}-2 y=0$ given $y=0, \frac{d y}{d x}=0, \frac{d^{2} y}{d x^{2}}=6$ when $x=0$ using Laplace transform.

Explanation

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