Laplace Transforms - Solving Initial Value Problems

You will learn how to solve second-order (and higher) initial value problems using Laplace transforms, including problems with step functions and more complex forcing terms.

Laplace transforms shine brightest on initial value problems. The process is:

1. Take the Laplace transform of every term in the differential equation.
2. Use the derivative rules to incorporate initial conditions automatically.
3. Solve the resulting algebraic equation for $Y(s)$.
4. Use partial fractions and inverse transforms to get back to $y(t)$.

This method is especially powerful because it handles initial conditions cleanly and works well with discontinuous forcing functions.

The visual shows the solution to $y'' + 4y' + 5y = 0$ with $y(0) = 1$, $y'(0) = -2$. The result is a damped oscillation $y = e^{-2t}(\cos t - \sin t)$, obtained via Laplace transforms. The dashed envelope shows the decay rate.

Example 1: Standard Second-Order IVP

Solve $y'' + 4y' + 5y = 0$ with $y(0) = 1$, $y'(0) = -2$.

Solution:

$$s^2 Y - s + 2 + 4(sY - 1) + 5Y = 0$$ $$(s^2 + 4s + 5)Y = s + 2$$ $$Y(s) = \frac{s + 2}{s^2 + 4s + 5} = \frac{s + 2}{(s+2)^2 + 1}$$

Complete the square and recognize the form:

$$y(t) = e^{-2t}\cos t$$

Wait, let’s be more careful. $Y = \frac{s+2}{(s+2)^2 + 1}$. Using the shift property: $y(t) = e^{-2t}\cos t$.

Example 2: With Forcing Function

Solve $y'' + y = \sin t$ with $y(0) = 0$, $y'(0) = 1$.

Solution:

$$s^2 Y - 1 + Y = \frac{1}{s^2 + 1}$$ $$Y(s) = \frac{1}{s^2+1} + \frac{1}{(s^2+1)^2}$$

The inverse gives: $y(t) = \sin t + \frac{1}{2}(\sin t - t\cos t)$

Example 3: Step Function Input

Solve $y'' + 4y = u(t - \pi)$ (unit step at $t = \pi$) with zero initial conditions.

Solution: The Laplace transform of the step function gives $\frac{e^{-\pi s}}{s}$:

$$Y(s) = \frac{e^{-\pi s}}{s(s^2 + 4)}$$

Using partial fractions and the time-shift theorem:

$$y(t) = u(t-\pi) \cdot \frac{1}{4}\left(1 - \cos 2(t-\pi)\right)$$

The solution is zero until $t = \pi$, then begins oscillating.

Engineers use this technique constantly to analyze systems with sudden inputs: a voltage spike in a circuit, a sudden force on a mechanical system, or a control signal sent to a motor. In control theory, Laplace transforms let designers predict how a system will respond to real-world commands and disturbances before building the physical prototype.