17calculus 17calculus
First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

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Solve Differential Equations Using Laplace Transforms

If you haven't seen Laplace Transforms before, the calculus section of 17calculus has a complete discussion of what they are and how to work with them.

Using Laplace Transforms to solve differential equation initial value problems is a great way to streamline solutions and, for forcing functions that are discontinuous, they are about the only way to solve them. The equations we use are

Laplace Transforms of Derivatives

\(\displaystyle{ \mathcal{L}\{y'\} = sY(s) - y(0) }\)

\(\displaystyle{ \mathcal{L}\{y''\} = s^2Y(s) - sy(0) - y'(0) }\)

where \(\mathcal{L}\{y(t)\} = Y(s)\)

\(\displaystyle{ \mathcal{L}\{ f^{(n)}(t) \} = }\) \(\displaystyle{ s^nF(s) - s^{n-1}f(0) - }\) \(\displaystyle{ s^{n-2}f'(0) - . . . - f^{(n-1)}(0) }\)

The idea is that we are given a differential equation which we first transform using the Laplace Transform, we solve it in the s-domain and then perform an inverse Laplace transform back to the t-domain to get our answer. Here is a very quick video giving these steps in a flow-chart.

PatrickJMT - Laplace Transform flow-chart

This video clip explains in detail how to solve differential equations using Laplace Transforms including a quick calculation of the Laplace transform of the first derivative function above.

Dr Chris Tisdell - solve differential equations using Laplace Transforms

This video expands on the one above taking you step-by-step through this process and, at the end of the video, gives you a big picture of how to do this. This is a great video.

Dr Chris Tisdell - big picture

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Practice Problems

Instructions - - Unless otherwise instructed, solve these initial value problems using Laplace transforms. Give your answers in exact, completely factored form.
In the following problems, \(u(t)\) is the unit step function (Heaviside function).

Level A - Basic

Practice A01

\(y'+y=u(t-1)\); \(y(0)=0\)

solution

Practice A02

\(\displaystyle{y^{(4)}-y=0}\); \(y(0)=0\), \(y'(0)=1\), \(y''(0)=0\), \(y'''(0)=0\)

solution


Level B - Intermediate

Practice B01

\(y''+4y=g(t)\); \(y(0)=y'(0)=0\)

answer

solution

Practice B02

\(y''-3y'+2y=g(t)\); \(y(0)=1, y'(0)=3\)

answer

solution

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