Differential Equations Answers

Find the Laplace transform Y(s)=L{y} of the solution of the given initial value problem.

y′′ + 4y = { t, 0 ≤ t < 1
{ 1, 1 ≤ t < ∞, y(0) = 3, y′(0) = 3


Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).

Y(s)= _______________

Find the Laplace transform Y(s)=L{y} of the solution of the given initial value problem.

y′′ +16y = {1, 0 ≤t < π
{0, π ≤ t < ∞, y(0) = 3, y′(0) = 5

Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).

Y(s) = ___________

Use the Laplace transform to solve the given initial value problem.

y′′−4y′−96y=0; y(0)=14, y′(0)= 28


Enclose arguments of functions in parentheses. For example, sin(2x).

y =___________

Find the inverse Laplace transform L−1{F(s)} of the given function.

F(s) = 6s^2−10s+16 / s(s^2+4)

Your answer should be a function of t. Enclose arguments of functions in parentheses. For example, sin(2x).

L^−1 {F(s)} = _______________-

Using integration by parts, find the Laplace transform of the given function; a is a real constant.

f(t) = t^2 sinh(at)

Your answer should be an expression in terms of a and s.
Enclose numerators and denominators in parentheses. For example, (a−b)/(1+n).


L{f(t)}(s )= F(s )= ______________

Find an explicit solution of the given initial-value problem

dy/dx = ye^(-x^2), y(5) = 1

Find an explicit solution of the given initial- value problem

square root of 1-y^2 dx - square root 1-x^2 dy = 0, y(0)= 1/2

y = _________________

Solve the given initial-value problem

(3y + 2t - 5)dt + (4y+ 3t - 1)dy = 0, y(-1)=2

(2x+1)(x+1)d^2y/dx^2+2xdy/dx-2y=(2x+1)^2

Find ℒ{f(t)}

by first using a trigonometric identity. (Write your answer as a function of s.)
f(t) = sin(5t + 6)