Solving DEs with Laplace Transforms Classroom

99.1 Introduction to Solving DEs with Laplace Transforms

Laplace Transforms provide a powerful method for solving linear ordinary differential equations (ODEs), especially those with constant coefficients and initial conditions. The key advantage is that they convert a differential equation in the time domain into an algebraic equation in the complex frequency domain (s-domain).

This transformation simplifies the problem significantly, allowing us to use algebraic techniques to solve for the transformed variable, and then use the inverse Laplace Transform to obtain the solution in the original time domain.

99.2 Procedure for Solving DEs with Laplace Transforms

The general procedure for solving an ODE using Laplace Transforms involves four main steps:

1. Transform the ODE: Take the Laplace Transform of both sides of the differential equation. Use the linearity property and the transform formulas for derivatives (e.g., $\mathcal{L}\{y'(t)\} = sY(s) - y(0)$ and $\mathcal{L}\{y''(t)\} = s^2Y(s) - sy(0) - y'(0)$) and any forcing functions. This converts the ODE into an algebraic equation in terms of $Y(s)$, the Laplace Transform of the solution $y(t)$.
2. Solve for $Y(s)$: Algebraically rearrange the transformed equation to isolate $Y(s)$. This often involves factoring and simplifying the expression for $Y(s)$.
3. Perform Partial Fraction Decomposition (if necessary): If $Y(s)$ is a complex rational function, decompose it into simpler fractions using partial fraction decomposition. This prepares $Y(s)$ for the inverse Laplace Transform.
4. Inverse Transform $Y(s)$: Take the inverse Laplace Transform of $Y(s)$ to find the solution $y(t)$ in the time domain. Use tables of inverse Laplace Transforms and properties like the Second Shifting Theorem for terms involving $e^{-cs}$.

99.3 Solving DE Problems

Let's apply the procedure to solve a first-order differential equation with an initial condition.

100.1 Introduction to Solving Simultaneous DEs (Laplace)

The Laplace Transform method can be extended to solve systems of linear ordinary differential equations, which often arise in coupled physical systems (e.g., connected mass-spring systems, multi-loop electrical circuits). Just as with single DEs, the Laplace Transform converts the system of differential equations into a system of algebraic equations in the s-domain.

This allows us to use standard algebraic techniques (like substitution or Cramer's rule) to solve for the transformed variables, and then inverse transform them back to the time domain to find the individual solutions for each dependent variable.

100.2 Procedure for Solving Simultaneous DEs (Laplace)

The steps for solving simultaneous DEs are a direct extension of solving single DEs:

1. Transform Each Equation: Take the Laplace Transform of each differential equation in the system. Apply the transform formulas for derivatives and initial conditions to convert each DE into an algebraic equation in the s-domain.
2. Form a System of Algebraic Equations: You will now have a system of linear algebraic equations involving $Y_1(s), Y_2(s), \dots$ (the Laplace Transforms of your dependent variables).
3. Solve the Algebraic System: Use algebraic methods (e.g., substitution, elimination, Cramer's rule, matrix methods) to solve for each transformed variable $Y_i(s)$ individually.
4. Perform Partial Fraction Decomposition (if necessary): For each $Y_i(s)$, if it's a complex rational function, decompose it into simpler fractions.
5. Inverse Transform Each $Y_i(s)$: Take the inverse Laplace Transform of each $Y_i(s)$ to find the solutions $y_i(t)$ in the time domain.

100.3 Solving Simultaneous DE Problems

Let's work through an example of solving a simple system of simultaneous differential equations using Laplace Transforms.