- #1

- 351

- 87

- Homework Statement
- Nil

- Relevant Equations
- Nil

We have to solve

$$

\begin{align*}

y'' - 5y' - 6y = e^{3x} \\

y(0) = 2,~~ y'(0) = 1 \\

\end{align*}

$$

Applying Laplace Transform the equation

$$

\begin{align*}

L [ y''] - 5 L[y'] - 6 L[y] = L [ e^{3x} ] \\

s^2 Y(s) - \left( s y(0) + y'(0) \right) - 5s Y(s) + y(0) - 6 Y(s) = \frac{1}{s-3} \\

Y(s) \{ s^2 - 5s - 6\} = \frac{1}{s-3} + 2s - 9 \\

Y(s) = \frac{2s^2 - 15s +27}{(s-3) (s+1) (s-6)} \\

\textrm{On Partial Fraction Decomposition}\\

\frac{A}{s-3} + \frac{B}{s+1} + \frac{C}{s-6} = \frac{2s^2 -15s +27}{(s-3) (s+1) (s-6)} \\

A (s+1)(s-6) + B(s-3)(s-6) + C (s-3)(s+1) = 2s^2 -15s +27 \\

\textrm{putting s =6} \\

21C = 9 \implies C = \frac{3}{7} \\

\textrm{similarly,} \\

s = -1 ~\text{gives}~ B = 11/7 \\

s = 3 ~ \text{gives} ~ A = 0\\

\textrm{Thus,}~~ y_p = 11/7 L^{-1} [1/(s+1)] + 3/7 L^{-1} [1/(s-6)] \\

y_p = 11/7 e^{-x} + 3/7 e^{6x}

\end{align*}

$$

But this doesn't match with the answer given in the book. Where is my mistake?

$$

\begin{align*}

y'' - 5y' - 6y = e^{3x} \\

y(0) = 2,~~ y'(0) = 1 \\

\end{align*}

$$

Applying Laplace Transform the equation

$$

\begin{align*}

L [ y''] - 5 L[y'] - 6 L[y] = L [ e^{3x} ] \\

s^2 Y(s) - \left( s y(0) + y'(0) \right) - 5s Y(s) + y(0) - 6 Y(s) = \frac{1}{s-3} \\

Y(s) \{ s^2 - 5s - 6\} = \frac{1}{s-3} + 2s - 9 \\

Y(s) = \frac{2s^2 - 15s +27}{(s-3) (s+1) (s-6)} \\

\textrm{On Partial Fraction Decomposition}\\

\frac{A}{s-3} + \frac{B}{s+1} + \frac{C}{s-6} = \frac{2s^2 -15s +27}{(s-3) (s+1) (s-6)} \\

A (s+1)(s-6) + B(s-3)(s-6) + C (s-3)(s+1) = 2s^2 -15s +27 \\

\textrm{putting s =6} \\

21C = 9 \implies C = \frac{3}{7} \\

\textrm{similarly,} \\

s = -1 ~\text{gives}~ B = 11/7 \\

s = 3 ~ \text{gives} ~ A = 0\\

\textrm{Thus,}~~ y_p = 11/7 L^{-1} [1/(s+1)] + 3/7 L^{-1} [1/(s-6)] \\

y_p = 11/7 e^{-x} + 3/7 e^{6x}

\end{align*}

$$

But this doesn't match with the answer given in the book. Where is my mistake?