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Differential Equations - The Laplace Transform

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    Study Guide Differential Equations – The Laplace Transform 1. Solving Differential Equations The Laplace transform is a powerful tool for solving first - and second - order differential equations with constant coefficients , especially when the problem is an initial value problem ( IVP) . A key requirement is that the initial conditions are given at ( x = 0 ) . 1.1 Why Use Laplace Transforms? Normally, when solving an IVP: 1. You first find the general solution of the differential equation. 2. Then you use the initial conditions to determine the constants. The Laplace transform works differently — and often more efficiently: • It automatically builds the initial conditions into the solution process. • It turns a differential equation into an algebraic equation , which is usually easier to solve. 1.2 The Basic Strategy To solve an IVP using Laplace transforms, follow these steps: 1. Apply the Laplace transform ( L) to both sides of the differential equation. 2. Use properties of the Laplace transform to rewrite the equation in terms of ( L[y] ) . 3. Solve the resulting algebraic equation for ( L[y] ) . 4. Take the inverse Laplace transform to find ( y ( x) ) . 1.3 Laplace Transforms of Derivatives Before applying the method, we need to know how the Laplace transform acts on derivatives. First derivative Using integration by parts, we find:

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    Study Guide Second derivative Applying the same idea again: These formulas are essential because they allow us to replace derivatives with algebraic expressions involving initial values. 1.4 Common Laplace Transforms Many functions form closed families under differentiation. This means repeated derivatives stay within a small group of related functions. Some important examples: Knowing these families makes it easier to recognize inverse transforms later. Example 1 : First - Order IVP Solve:

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    Study Guide Step 1: Apply the Laplace transform Using the derivative rule and initial condition: Substitute ( y ( 0) = - 5 ) : Step 2: Solve for ( L [ y] ) Step 3: Take the inverse Laplace transform Using partial fractions: Taking inverse transforms: Example 2 : Second - Order IVP Solve:

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    Study Guide Step 1: Apply the Laplace transform Substitute initial values: Step 2: Solve After partial fraction decomposition and inversion: Example 3 : Use Laplace Transforms to Solve the IVP with initial conditions: We are going to solve this step by step using Laplace transforms . The big idea is: • Take the Laplace transform of both sides. • Use the initial conditions. • Solve for ( L[y] ) . • Take the inverse Laplace transform to find ( y ( t) ) .

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    Study Guide Step 1: Take the Laplace Transform Apply ( L) to both sides: Using linearity: Step 2: Use Laplace Formulas Recall: Now substitute the initial conditions: • ( y ( 0) = - 1 ) • ( y' ( 0) = 7 ) So: Step 3: Substitute Everything Back Now simplify: Group like terms:

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    Study Guide Step 4: Solve for ( L[y]) Step 5: Prepare for Inverse Laplace Rewrite the denominator: Complete the square: So: Rewrite the numerator: So: Rewrite the second fraction to match a standard form:

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    Study Guide Step 6: Take the Inverse Laplace Transform Use standard formulas: Here: • ( a = 1 ) • ( b = 2 ) So: Final Answer What to Notice • We used Laplace transforms to turn a differential equation into algebra. • Initial conditions were inserted early in the process. • Completing the square helped match standard Laplace formulas. • The final solution has the form ( e at ( A cos bt + B sin bt) ) , which is typical for complex roots. Example 4 : Step Functions and Discontinuous Inputs The Laplace transform is especially useful when the forcing function is discontinuous , such as a step function .

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    Study Guide Figure 1 Figure 2 Consider: where

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    Study Guide Key Laplace property If then for any positive constant ( k ) , Applying the method After transforming and solving: Using inverse transforms and the step - function property: This solution matches the expected behavior : nothing happens before ( x = 2 ) , and the solution changes afterward. Summary • Laplace transforms turn differential equations into algebraic equations. • Initial conditions are built directly into the method. • The approach works especially well for IVPs with constant coefficients. • Step functions and discontinuities are handled naturally using exponential shifts. • Inverse transforms rely heavily on recognizing known function families.

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