Learn Laplace Transform Equations with Plainmath

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

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

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 ${\displaystyle \frac{dy}{dx}=4}$?

${\displaystyle d^2\frac{y}{{dx}^2}-2\frac{dy}{dx}+10y=0}$ 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.

Inverse Laplace transformation

$(s^2+s)/(s^2+1)(s^2+2s+2)$