# Laplace transform differential equation examples

Using the Laplace transform solve mx ″ + cx ′ + kx = 0, x(0) = a, x ′ (0) = b. where m > 0, c > 0, k > 0, and c2 = 4km (system is critically damped). Exercise 6.E. 6.2.6 Solve x ″ + x = u(t − 1) for initial conditions x(0) = 0 and x ′ (0) = 0. Exercise 6.E. 6.2.7 Show the differentiation of the transform property. Suppose L{f(t)} = F(s), then show

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## 16 Laplace transform. Solving linear ODE

Using the properties of the Laplace transform, we can transform this constant coefficient differential equation into an algebraic equation. s 2 Y − s y ( 0 ) − y ′ ( 0 ) + 5 ( s Y − y ( 0 ) ) + 6 Y = s s 2 + 1 s 2 Y − s + 5 s Y − 5 + 6 Y = s s 2 + 1 {\displaystyle {\begin{aligned}s^{2}Y
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