Solve System of Differential Equations with Expert Help

Find the Laplace transform of ${\displaystyle {\cos }^{2}t}$

use the inverse form of the First translation theorem to find the following laplace transform
$L^{-1}\left\{\frac{s}{s^2+4s+5}\right\}$

Solve the following initial value problem by using Laplace transform
$y^{″}+4y=4t+12,y(0)=4,y^{\prime }(0)=0$

Solve the following differential equation by the laplace transform method
${\displaystyle 6y^{\left(4\right)}+7y{}^{‴}+21y28y^{\prime }-12y=t(\cos \left(2t\right)+t{e}^{\frac{-3t}{2}})}$
$y(0)=0,y^{\prime }(0)=1,y"(0)=0,y^{‴}(0)=-1$

Solve the differential equation using either the method of undetermined coefficients or Laplace transforms.
${\displaystyle y-6ytext11y-6y=3x}$

Solve the following differential equations using the Laplace transform
$y"-4y=0$
$y(0)=0,y^{\prime }(0)=0$

Find the Laplace transform of ${\displaystyle \sin ht-\cos ht}$

Evaluate the laplace transform for this
a)$L\{4x^2-3\cos (2x)+5e^x\}$
b)$L\{4+\frac{1}{2}\sin (x)-e^{4x}\}$

Determine the Laplace transform of the following integrals
$L\left\{{\int }_0^t{e}^{-2t}{t}^{2}dt\right\}$

Solve ${\displaystyle y^{\prime }+y=4\delta (t-2\pi )}$
Subject to
y(0)=0
y'(0)=0

Use the Laplace transform to solve the given initial-value problem.
${\displaystyle \frac{dy}{dt}-y=z,y\left(0\right)=0}$

Can you indefinite the integral of ${\displaystyle x^2\arcsin \left(x\right)}$ please?

${\displaystyle \frac{d^{2}y}{{dx}^2}-2\frac{dy}{dx}+10y=0}$ where x=0.y=0 and ${\displaystyle \frac{dy}{dx}=4?x=0.y=0}$ and ${\displaystyle \frac{dy}{dx}=4}$?

We give linear equations y=1.5x

We give linear equations y=-3x

Write first-order differential equations that express the situation: The increase in the number of people who have heard a rumor in a town of population 20,00020,000 is proportional to the product of the number PP who have heard the rumor and the number who have not heard the rumor at time tt .

${\displaystyle y^{\prime }={(y+4x)}^2}$

${\displaystyle d^2\frac{y}{{dx}^2}-2\frac{dy}{dx}+10y=0}$ where $x=0;y=0$ and ${\displaystyle \frac{dy}{dx}=4?x=0;y=0}$ and ${\displaystyle \frac{dy}{dx}=4}$?

In many physical applications, the nonhomogeneous term F(x) is specified by different formulas in different intervals of x. (a) Find a general solution of the equation ${\displaystyle y+y=\{x,0\le x\le 1,1\le x\}}$ NoteNote that the solution is not differentiable at x = 1. (b) Find a particular solution of $y+y=\{x,0\le x\le 1;1,1\le x\}$ that satises the initial conditions y(0)=0 and y′(0)=1.

${\displaystyle 2\cdot \left(\frac{dy}{dx}\right)+2y=0}$