Solve System of Differential Equations with Expert Help

solve the given initial value problem
${\displaystyle y{}^{‴}+3y{}^{″}-13y^{\prime }-15y=0,y\left(0\right)=y^{\prime }\left(0\right)=0,y{}^{″}\left(0\right)=1}$

Find the solution of the given system of differential equations by Operator method or Laplace transform.
${\displaystyle 2\frac{dx}{dt}+\frac{dy}{dt}+x+5y=4t}$
${\displaystyle \frac{dx}{dt}+\frac{dy}{dt}+2x+2y=2}$
${\displaystyle x\left(0\right)=3,y\left(0\right)=-4}$

Use Laplace transforms to solve the initial value problems.
$y"+4y=2\delta (t-\pi ),y(0)=1,y^{\prime }(0)=0$

Find laplace transform of the following functions
${\displaystyle f\left(t\right)=t\sin h\left(2t\right)}$

Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by
${\displaystyle F\left(s\right)={\int }_0^{\mathrm{\infty }}{e}^{-st}f\left(t\right)dt}$
where we assume s is a positive real number. For example, to find the Laplace transform of ${\displaystyle f\left(t\right)=e^{-t}}$, the following improper integral is evaluated using integration by parts:
${\displaystyle F\left(s\right)={\int }_0^{\mathrm{\infty }}{e}^{-st}{e}^{-t}dt={\int }_0^{\mathrm{\infty }}{e}^{-(s+1)t}dt=\frac{1}{s+1}}$
Verify the following Laplace transforms, where u is a real number.
${\displaystyle f\left(t\right)=\cos at\to F\left(s\right)=\frac{s}{s^2+a^2}}$

Consider the following initial value problem:
${\displaystyle y\prime \prime -4y\prime -45y=\sin \left(7t\right),y\left(0\right)=1,y\prime \left(0\right)=-4}$
Using Y for the Laplace transform of y(t), i.e., Y=L{y(t)}, find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)=?

Solve the differential equation by variation of parameters
${\displaystyle yy=\sin x}$

Find the initial value for the function $f\left\{t\right\}=2u(t)+3\cos u(t)$
a) 2
b) 3
c) 5
d) 4

Calculate the Laplace transform $L\left\{f\right\}$ of the function ${\displaystyle f\left(t\right)=e^{4t+2}-8t^6+8\sin \left(2t\right)+9}$ using the basic formulas and the linearity of the Laplace transform

Find the Laplace transform by the method of definition.
${\displaystyle f\left(t\right)=e^{\frac{t}{5}}}$

Solve the following differential equations using the Laplace transform
${\displaystyle y2y^{\prime }+y=t+3}$
$y(0)=1,y^{\prime }(0)=0$

Evaluate ${\displaystyle L^{-1}\left(\frac{3x}{x^2+9}\right)}$ where ${\displaystyle L^{-1}}$ is inverse Laplace transformation?
a)${\displaystyle 2\cos \left(2x\right)}$
b)${\displaystyle 3\sin \left(3x\right)}$
c)${\displaystyle 3\cos \left(9x\right)}$
d)${\displaystyle 3\cos \left(3x\right)}$
e)${\displaystyle 2\cos \left(3x\right)}$

Find the Laplace transform of ${\displaystyle 3e^{-7t}}$

Find the inverse Laplace transform of the following:
$\left(\frac{7}{2}\right)\log \left(\frac{s-8}{s+8}\right)$

Use the Laplace transform table and the linearity of the Laplace transform to determine the following transform
$L\{t^4+t^3-t^2-t+\sin \sqrt{2}t\}$

Find Laplace transform $L\{10e^{-5t}\sin \frac{t}{2}\}$

Which of the following is a TRUE statement?
A) $L^{-1}\left\{\frac{6}{s^7}\right\}=t^6$
B) $L^{-1}\left\{\frac{3}{s^2+s}\right\}=3-3e^t$
C) $L^{-1}\left\{\frac{6!}{(s+4{)}^8}\right\}=t^6{e}^{4t}$
D) $L^{-1}\left\{\frac{s}{s^2+5s+6}\right\}=3e^{-3t}-2e^{-2t}$
E) $L^{-1}\left\{\frac{5!}{(s+2{)}^3}\right\}=t^5{e}^{-2t}$

Invert the following system to the time domain using partial fraction
$f(s)=\frac{6}{s^2(s+2)(s+4)}$

Determine the solution of y(t) for the following equations
${\displaystyle \frac{dy}{dt}+5{\int }_0^{t}ydt+2y=10\text{ with }y\left(0\right)=0}$

If the $L\left\{y\right\}=\bar{y},\text{ ,then the }L\left\{y"\right\}$ is
a) ${\displaystyle s\bar{y}-y\left(0\right)}$
b) ${\displaystyle s^2\bar{y}-sy\left(0\right)-y^{\prime }\left(0\right)}$
c) ${\displaystyle s^2\bar{y}+sy\left(0\right)+y^{\prime }\left(0\right)}$
d) None of these