Question

Differential Equations

3a. Use the integral definition of the Laplace transform to show that Carefully indicate the range of s for which the Laplace transform is defined, and clearly show where you use this in your calculations.

b. Calculate the convolution

Answer #1

Topic: Differential Equations, Laplace Transform, particular
solutions
Please explain what is a Laplace transform, why it works, and
why you must use the integral of e^-st, rather than the Laplace
shortcuts, when solving for a piecewise function (with 2+
equations).

Use Definition 7.1.1,
DEFINITION 7.1.1 Laplace
Transform
Let f be a function defined for
t ≥ 0.
Then the integralℒ{f(t)} =
∞
e−stf(t) dt
0
is said to be the Laplace transform of
f, provided that the integral converges.
to find
ℒ{f(t)}.
(Write your answer as a function of s.)
f(t) = te8t
ℒ{f(t)} =
(s > 8)

Given use Laplace transform to solve the following systems of
differential equations.
2x' - y' - z' = 0
x' + y' = 4t + 2
y' + z = t2 + 2
SUBJECT = ORDINARY DIFFERENTIAL EQUATIONS
TOPIC = LAPLACE TRANSFORM

Use the Laplace transform to solve the given system of
differential equations. dx dt = −x + y dy dt = 2x x(0) = 0, y(0) =
2

Use the Laplace transform to solve the given system of
differential equations. d2x dt2 + d2y dt2 = t2 d2x dt2 − d2y dt2 =
3t x(0) = 8, x'(0) = 0, y(0) = 0, y'(0) = 0

Use the definition of the Laplace transform to find the
transform of:
f(t)= at+ b cos(2t), where a, b are real numbers.
Please solve using simple and easy to follow steps (here for
learning not for answers)
Thanks

Differential Equations: Use the Laplace transform to solve the
given initial value problem:
y′′ −2y′ +2y=cost;
y(0)=1,
y′(0)=0

Use the Laplace transform to solve the given system of
differential equations. 2 dx/dt + dy/dt − 2x = 1
dx/dt + dy/dt − 6x − 6y = 2
x(0) = 0, y(0) = 0

Use the Laplace transform to solve the given system of
differential equations. dx/dt=x-2y dy/dt=5x-y x(0) = -1, y(0) =
6

Use the Laplace Transform method to solve the following
differential equation problem: y 00(t) − y(t) = t + sin(t), y(0) =
0, y0 (0) = 1
Please show partial fraction steps to calculate
coeffiecients.

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