# How to solve for third order differential equation of y"'-7y'+6y =2 sin (t) using Method of Laplace Transform when y(0)=0, y'(0)=0, y"(0)=0? Step by step

Question
Laplace transform
How to solve for third order differential equation of $$y"'-7y'+6y =2 \sin (t)$$ using Method of Laplace Transform when $$y(0)=0, y'(0)=0, y"(0)=0$$?
Step by step

2021-03-10
Step 1
Given third order differential equation is: $$y"'-7y'+6y =2 \sin (t)$$
Apply Laplace transform on each term on both sides of the differential equation.
$$L\left\{y'''\right\}-7L\left\{y'\right\}+6L\left\{y\right\}=2L\left\{\sin t\right\}$$
Use the standard appropriate Laplace transforms for the higher order differentials:
$$L\left\{y'''\right\}=s^3Y(s)-s^2y(0)-sy'(0)-y''(0)$$
$$L\left\{y'\right\}=sY(s)-y(0)$$
$$L\left\{y\right\}=Y(s)$$ The Laplace transform of $$\sin(t)$$ is: $$L\left\{\sin t\right\}=\frac{1}{(1+s^2)}$$
Substitute the formula in the given equation:
$$\left[s^3Y(s)-s^2y(0)-sy'(0)-y''(0)\right]-7\left[sY(s)-y(0)\right]+6\left[Y(s)\right]=\frac{2}{(1+s^2)}$$
Now, substitute the given boundary conditions in the above equation.
$$\Rightarrow (s^3-7s+6)Y(s)=\frac{2}{(1+s^2)}$$
$$\Rightarrow Y(s)=\frac{2}{(1+s^2)(s^3-7s+6)}$$
Here, the denominator on the right hand side is factorized into:
$$\Rightarrow Y(s)=\frac{2}{(1+s^2)(s-1)(s-2)(s+3)}$$
Step 2 Now, find the inverse Laplace function of Y(s) to get the solution of the differential equation in the form of y(t)
$$Y(s)=\frac{2}{(1+s^2)(s-1)(s-2)(s+3)}$$
For this equation evaluate the partial fractions:
$$\frac{2}{(1+s^2)(s-1)(s-2)(s+3)}=\frac{A}{(s-1)}+\frac{B}{(s-2)}+\frac{C}{(s+3)}+\frac{D}{(1+s^2)}$$
$$\Rightarrow 2=A(1+s^2)(s−2)(s+3)+B(1+s^2)(s−1)(s+3)+C(1+s^2)(s−1)(s−2)+D(s−1)(s−2)(s+3)$$
$$\text{For }s=1, \text{value of A is: }A=-\frac{1}{5}$$
$$\text{For }s=2, \text{value of B is: }B=\frac{2}{25}$$
$$\text{For }s=-3, \text{value of C is: }C=\frac{1}{100}$$
Equating constant terms o both sides gives D value as: $$D=\frac[17}{100}$$
Then, the Laplace equation turns out to be:
$$Y(s)=\frac{17}{100}\left(\frac{1}{1+s^2}\right)-\frac{1}{5}\left(\frac{1}{s-1}\right)+\frac{2}{25}\left(\frac{1}{s-2}\right)+\frac{1}{100}\left(\frac{1}{s+3}\right)$$
Now, apply inverse Laplace on each term of the equation. $$y(t)=\frac{17}{100}L^{-1}\left\{\frac{1}{1+s^2}\right\}-\frac{1}{5}L^{-1}\left\{\frac{1}{s-1}\right\}+\frac{2}{25}L^{-1}\left\{\frac{1}{s-2}\right\}+\frac{1}{100}L^{-1}\left\{\frac{1}{s+3}\right\}$$
Use the standard Laplace transforms: $$L^{-1}\left\{\frac{1}{(s-a)}\right\}=e^{at} \text{ and } L^{-1}\left\{\frac{a}{(a+s^2)}\right\}=\sin(at)$$
Which gives the solution as:
$$y(t)=\frac{17}{100}\sin t-\frac{1}{5}e^t+\frac{2}{25}e^{2t}+\frac{1}{100}e^{-3t}$$

### Relevant Questions

ALSO, USE PARTIAL FRACTION WHEN YOU ARRIVE
$$L(y) = \left[\frac{w}{(s^2 + a^2)(s^2+w^2)}\right]*b$$
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$$\frac{d^2y}{dt^2}+a^2y=b \sin(\omega t)$$ where $$y(0)=0$$
and $$y'(0)=0$$
Use the Laplace transform to solve the given differential equation subject to the indicated initial conditions.
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a) $$\frac{14}{{20}}{e}^{{{2}{t}}}-\frac{5}{{30}}{e}^{{-{2}{t}}}-\frac{9}{{30}}{e}^{{-{6}{t}}}$$
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c) $$\frac{14}{{15}}{e}^{{{2}{t}}}-\frac{5}{{10}}{e}^{{-{2}{t}}}-\frac{9}{{20}}{e}^{{-{3}{t}}}$$
d) $$\frac{14}{{20}}{e}^{{{2}{t}}}+\frac{5}{{20}}{e}^{{-{2}{t}}}-\frac{9}{{20}}{e}^{{-{3}{t}}}$$
$$y'' + 6y' + 5y = t \cdot U (t-2)$$
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$$y'''+3y''−18y'−40y=−120$$
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