Question

Find the solution of the initial value problem given below by Laplace transform y'-y=t e^t sin t y(0)=0

Laplace transform
ANSWERED
asked 2021-03-06
Find the solution of the initial value problem given below by Laplace transform
\(y'-y=t e^t \sin t\)
\(y(0)=0\)

Answers (1)

2021-03-07
Step 1
The answer is given by using properties of Laplace transform.
Step 2
\(y'-y=t e^t \sin(t)\)
\(y(0)=0\)
Apply the laplace transform
\(L\left\{y'\right\}-L\left\{y\right\}=L\left\{t e^t \sin(t)\right\}\)
\(\Rightarrow (sY(s)-y(0))-Y(s)=\frac{2(s-1)}{(s^2-2s+2)^2}\)
\(\Rightarrow sY(s)-Y(s)=\frac{2(s-1)}{(s^2-2s+2)^2}\)
\(\Rightarrow (s-1)Y(s)=\frac{2(s-1)}{(s^2-2s+2)^2}\)
\(\Rightarrow Y(s)=\frac{2}{(s^2-2s+2)^2}\)
Now apply inverse laplace transform
\(\Rightarrow L^{-1}\left\{Y(s)\right\}=L^{-1}\left\{\frac{2}{(s^2-2s+2)^2}\right\}\)
\(\Rightarrow y(t)=L^{-1}\left\{\frac{2}{((s-1)^2+1)^2}\right\}\)
\(\Rightarrow y(t)=e^t \sin(t)-t e^t \cos(t)\)
\(\Rightarrow y(t)=e^t(\sin(t)-t\cos(t))\)
0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-05-16
Use the Laplace transform to solve the given initial-value problem.
\(dy/dt-y=z,\ y(0)=0\)
asked 2020-12-25

Let x(t) be the solution of the initial-value problem
(a) Find the Laplace transform F(s) of the forcing f(t).
(b) Find the Laplace transform X(s) of the solution x(t).
\(x"+8x'+20x=f(t)\)
\(x(0)=-3\)
\(x'(0)=5\)
\(\text{where the forcing } f(t) \text{ is given by }\)
\(f(t) = \begin{cases} t^2 & \quad \text{for } 0\leq t<2 ,\\ 4e^{2-t} & \quad \text{for } 2\leq t < \infty . \end{cases}\)

asked 2021-02-19

Use Laplace transform to solve the following initial-value problem
\(y"+2y'+y=0\)
\(y(0)=1, y'(0)=1\)
a) \(\displaystyle{e}^{{-{t}}}+{t}{e}^{{-{t}}}\)
b) \(\displaystyle{e}^{t}+{2}{t}{e}^{t}\)
c) \(\displaystyle{e}^{{-{t}}}+{2}{t}{e}^{t}\)
d) \(\displaystyle{e}^{{-{t}}}+{2}{t}{e}^{{-{t}}}\)
e) \(\displaystyle{2}{e}^{{-{t}}}+{2}{t}{e}^{{-{t}}}\)
f) Non of the above

asked 2021-03-07

use the Laplace transform to solve the initial value problem.
\(y"-3y'+2y=\begin{cases}0&0\leq t<1\\1&1\leq t<2\\ -1&t\geq2\end{cases}\)
\(y(0)=-3\)
\(y'(0)=1\)

asked 2021-06-06
Use the table of Laplace transform and properties to obtain the Laplace transform of the following functions. Specify which transform pair or property is used and write in the simplest form.
a) \(x(t)=\cos(3t)\)
b)\(y(t)=t \cos(3t)\)
c) \(z(t)=e^{-2t}\left[t \cos (3t)\right]\)
d) \(x(t)=3 \cos(2t)+5 \sin(8t)\)
e) \(y(t)=t^3+3t^2\)
f) \(z(t)=t^4e^{-2t}\)
asked 2020-11-02
\(y''+y=e^{-2t}\sin t , y(0)=y′(0)=0\)
solution of given initial value problem with Laplace transform
asked 2020-12-24
Find the solution of the initial value problem given below by Laplace transform.
\(x"+16x=\cos(4t)\)
\(x(0)=0 \text{ and } x'(0)=1\)
asked 2021-02-16

Solution of I.V.P for harmonic oscillator with driving force is given by Inverse Laplace transform
\(y"+\omega^{2}y=\sin \gamma t , y(0)=0,y'(0)=0\)
1) \(y(t)=L^{-1}\bigg(\frac{\gamma}{(s^{2}+\omega^{2})^{2}}\bigg)\)
2) \(y(t)=L^{-1}\bigg(\frac{\gamma}{s^{2}+\omega^{2}}\bigg)\)
3) \(y(t)=L^{-1}\bigg(\frac{\gamma}{(s^{2}+\gamma^{2})^{2}}\bigg)\)
4) \( y(t)=L^{-1}\bigg(\frac{\gamma}{(s^{2}+\gamma^{2})(s^{2}+\omega^{2})}\bigg)\)

asked 2021-05-16
Find the Laplace transform of the function \(L\left\{f^{(9)}(t)\right\}\)
asked 2021-02-09
In an integro-differential equation, the unknown dependent variable y appears within an integral, and its derivative \(\frac{dy}{dt}\) also appears. Consider the following initial value problem, defined for t > 0:
\(\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}+{4}{\int_{{0}}^{{t}}}{y}{\left({t}-{w}\right)}{e}^{{-{4}{w}}}{d}{w}={3},{y}{\left({0}\right)}={0}\)
a) Use convolution and Laplace transforms to find the Laplace transform of the solution.
\({Y}{\left({s}\right)}={L}{\left\lbrace{y}{\left({t}\right)}\right)}{\rbrace}-?\)
b) Obtain the solution y(t).
y(t) - ?
...