Solving ODEs with the Laplace Transform
Transforming an ODE turns d/dt into multiplication by s with initial conditions absorbed automatically (L{y'} = sY - y(0)), converting the differential equation into algebra in Y(s); partial fractions and inverse transforms recover y(t). The method shines for piecewise and impulsive forcing, where step and delta functions transform into simple exponential factors.
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Transforming an ODE turns d/dt into multiplication by s with initial conditions absorbed automatically (L{y'} = sY - y(0)), converting the differential equation into algebra in Y(s); partial fractions and inverse transforms recover y(t). The method shines for piecewise and impulsive forcing, where step and delta functions transform into simple exponential factors.
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