Using Laplace transform, solve the equation:
given that:
Take the Laplace transform in $t$: $U(x,s)=\mathcal L\{u(x,t)\}$. Using $\mathcal L\{u_{tt}\}=s^2U-s\,u(x,0)-u_t(x,0)$ and $u_t(x,0)=0,\;u(x,0)=f(x)=10\sin(2\pi x)-20\sin(5\pi x)$, the PDE becomes $$ s^2U-sf(x)=9U_{xx}\quad\Longrightarrow\quad 9U_{xx}-s^2U=-s f(x). $$ Solve this ODE in $x$. Because $f$ is a ________ ___ _________ ______ _______ ________ _____ ___ _______ _____ _____ ______.
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Find:
Solve the boundary value problem:
y′′ − 3y′ + 2y = 2
with
y(0) - y'(0) = -1
y(1) + y'(1) = 1
using the second order finite difference method with step length
Solve the initial value problem y' = x2 + y2 y(0) = 1, upto x = 2.0 using third order Taylor series method with h = 0.1.
Using second order finite difference method, solve the boundary value problem:
Using step length
Solve the differential equation:
x2y" +6xy'+(6+x2)y=0
in series about x = 0.
Find the solution of = 0 in R subject to the boundary conditions:
where R is the square 0≤ x ≤1,0≤ y ≤1, using the five point formula. Use central difference approximation in the boundary condition. Assume uniform step length h = 1/2 along the axes.
Use Fourier transforms to solve the boundary value problem:
subject to the conditions:
i)
ii)