You CAN Ace Differential Equations

 basics of differential equations basics of laplace transforms convolution integral

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > differential equations > laplace transforms

If you haven't seen Laplace Transforms before, the Laplace Transform 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 [1min-33secs]

video by PatrickJMT

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 [28min-50secs]

video by Dr Chris Tisdell

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 [12min-45secs]

video by Dr Chris Tisdell

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Unless otherwise instructed, solve these initial value problems using Laplace transforms. Give your answers in exact, completely factored form.
Notes
1. In the following problems, $$u(t)$$ is the unit step function (Heaviside function).
2. Some of these problems may require convolution to solve.

Basic Problems

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

Problem Statement

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

Solution

### 646 solution video

video by Dr Chris Tisdell

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

Problem Statement

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

$$\displaystyle{ y(t) = \frac{1}{2}\left[ \sin(t) + \sinh(t) \right] }$$

Problem Statement

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

Solution

### 648 solution video

video by Dr Chris Tisdell

$$\displaystyle{ y(t) = \frac{1}{2}\left[ \sin(t) + \sinh(t) \right] }$$

Intermediate Problems

$$y''+4y=q(t), t > 0$$; $$y(0)=y'(0)=0$$ where $$\displaystyle{ q(t) = \left\{ \begin{array}{crl} t & & 0 \leq t \leq \pi/2 \\ \pi/2 & & t > \pi/2 \end{array} \right. }$$

Problem Statement

$$y''+4y=q(t), t > 0$$; $$y(0)=y'(0)=0$$ where $$\displaystyle{ q(t) = \left\{ \begin{array}{crl} t & & 0 \leq t \leq \pi/2 \\ \pi/2 & & t > \pi/2 \end{array} \right. }$$

$$\displaystyle{ y(t) = \frac{1}{4} \left[ t- \frac{1}{2} \sin(2t) \right] - \frac{u(t-\pi/2)}{4}\left[(t-\pi/2) - \frac{1}{2} \sin[2(t-\pi/2)] \right] }$$

Problem Statement

$$y''+4y=q(t), t > 0$$; $$y(0)=y'(0)=0$$ where $$\displaystyle{ q(t) = \left\{ \begin{array}{crl} t & & 0 \leq t \leq \pi/2 \\ \pi/2 & & t > \pi/2 \end{array} \right. }$$

Solution

### 647 solution video

video by Dr Chris Tisdell

$$\displaystyle{ y(t) = \frac{1}{4} \left[ t- \frac{1}{2} \sin(2t) \right] - \frac{u(t-\pi/2)}{4}\left[(t-\pi/2) - \frac{1}{2} \sin[2(t-\pi/2)] \right] }$$

$$y'' - 3y' + 2y = r(t)$$; $$y(0)=1, y'(0)=3$$ where $$r(t) = u(t-1) - u(t-2)$$

Problem Statement

$$y'' - 3y' + 2y = r(t)$$; $$y(0)=1, y'(0)=3$$ where $$r(t) = u(t-1) - u(t-2)$$

$$\displaystyle{ y(t) = -e^t + 2e^{2t} + u(t-1)\left[ \frac{1}{2} - e^{t-1} + \frac{1}{2}e^{2(t-1)} \right] - u(t-2)\left[ \frac{1}{2} - e^{t-2} + \frac{1}{2}e^{2(t-2)} \right] }$$

Problem Statement

$$y'' - 3y' + 2y = r(t)$$; $$y(0)=1, y'(0)=3$$ where $$r(t) = u(t-1) - u(t-2)$$

Solution

### 649 solution video

video by Dr Chris Tisdell

$$\displaystyle{ y(t) = -e^t + 2e^{2t} + u(t-1)\left[ \frac{1}{2} - e^{t-1} + \frac{1}{2}e^{2(t-1)} \right] - u(t-2)\left[ \frac{1}{2} - e^{t-2} + \frac{1}{2}e^{2(t-2)} \right] }$$

$$4y'' + y = g(t);$$ $$y(0)=3, y'(0)=-7$$

Problem Statement

$$4y'' + y = g(t);$$ $$y(0)=3, y'(0)=-7$$

$$y(t) = 3\cos(t/2) - 14\sin(t/2) + (1/2)\int_0^t{\sin(t/2)g(t-\tau)~d\tau}$$

Problem Statement

$$4y'' + y = g(t);$$ $$y(0)=3, y'(0)=-7$$

Solution

### 2242 solution video

video by Krista King Math

$$y(t) = 3\cos(t/2) - 14\sin(t/2) + (1/2)\int_0^t{\sin(t/2)g(t-\tau)~d\tau}$$

$$y'' + \omega_0^2 y = F_0 \sin(\omega t); y’(0)=0, y(0)=0$$

Problem Statement

$$y'' + \omega_0^2 y = F_0 \sin(\omega t); y’(0)=0, y(0)=0$$

$$\displaystyle{ y(t) = \frac{F_0}{\omega_0^2-\omega^2} \left[ \sin(\omega t) - \frac{\omega}{\omega_0}\sin(\omega_0 t) \right] }$$

Problem Statement

$$y'' + \omega_0^2 y = F_0 \sin(\omega t); y’(0)=0, y(0)=0$$

Solution

### 2259 solution video

video by Michel vanBiezen

$$\displaystyle{ y(t) = \frac{F_0}{\omega_0^2-\omega^2} \left[ \sin(\omega t) - \frac{\omega}{\omega_0}\sin(\omega_0 t) \right] }$$

$$y’ + 3y + 2\int_0^ty(\tau)~d\tau = 2e^{-3t}; y(0)=0$$

Problem Statement

$$y’ + 3y + 2\int_0^ty(\tau)~d\tau = 2e^{-3t}; y(0)=0$$

$$y(t) = e^{-3t}(-3+4e^t-e^{2t})$$

Problem Statement

$$y’ + 3y + 2\int_0^ty(\tau)~d\tau = 2e^{-3t}; y(0)=0$$

Solution

### 2261 solution video

video by Michel vanBiezen

$$y(t) = e^{-3t}(-3+4e^t-e^{2t})$$