Laplace transforms solve either homogeneous or nonhomogeneous linear systems in which the coefficients are all constants. Initial conditions appear explicitly in the solution.
Solve ODE + Laplace
= y, Laplace solution is: y
x
= y
0
ex
y′ + y = x + sin x (Specify x), Laplace solution is:
yx
= - 1 + x +
e-x + y
0
e-x +
sin x -
cos x
The following examples compare exact and Laplace solutions.
Equation | Exact | Laplace |
---|---|---|
y′ = sin x |
y![]() ![]() |
y![]() ![]() ![]() ![]() |
y′ = y + x |
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Dxy = cos x |
y![]() ![]() |
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Dxy = x + t |
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y![]() ![]() |
y![]() ![]() ![]() ![]() |
y′ = y2 + 1 |
-arctan![]() ![]() ![]() ![]() |
Fails |
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y![]() ![]() ![]() ![]() ![]() ![]() |
Fails |