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Unit 7: Nonhomogeneous Second-Order Equations

Forcing terms need a particular solution on top of the homogeneous one: guessing it with undetermined coefficients, or building it with variation of parameters.

8 topics · 28 questions

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3.5 The General Solution of a Nonhomogeneous Equation

  1. How it worksHow do the homogeneous solution and one particular solution add together to give every solution of the full equation?
  2. Worked exampleFind the general solution of y'' + y = x, given that yp = x is one particular solution.
  3. Common mistakeWhy isn't the particular solution yp expected to satisfy any initial conditions all by itself?
  4. Going furtherWhat happens to the general solution's structure when two different particular solutions are found by two different methods?
Your turn1 of 2
Given that yp = 3x is a particular solution of y'' + y = 3x, the general solution is y = C1 cos(x) + C2 sin(x) + 3x. Find y(0.61) when C1 = 3 and C2 = 4.

3.5 Undetermined Coefficients for Polynomial and Exponential Forcing

  1. How it worksHow do you match the form of a guessed particular solution to a polynomial or exponential forcing term?
  2. Worked exampleFind a particular solution of y'' - 3y' + 2y = 4x2 using undetermined coefficients.
  3. Worked exampleFind a particular solution of y'' + y' - 2y = 3e4x using undetermined coefficients.
Your turn1 of 2
Find the particular solution of y'' + 2y' + 4y = 4e6x using undetermined coefficients. Find yp(x).

3.5 Undetermined Coefficients for Sine and Cosine Forcing

  1. How it worksHow does guessing A cos(bt) plus B sin(bt) together handle a sinusoidal forcing term that neither piece alone could?
  2. Worked exampleFind a particular solution of y'' + 4y = 8 sin(3t) using undetermined coefficients.
  3. Going furtherWhat happens to the undetermined coefficients guess when the forcing term is cos(t) times an exponential?
  4. Worked exampleFind a particular solution of y'' - y = 2cos(t) + 3sin(t) using undetermined coefficients.
Your turn1 of 2
Find the particular solution yp(t) of y'' + 2.6272 y = 9sin(4.98t) using undetermined coefficients (w0 is not equal to w).

3.5 The Resonant Case in Undetermined Coefficients

  1. How it worksHow does multiplying the guessed particular solution by t rescue it when it duplicates a homogeneous term?
  2. Worked exampleFind a particular solution of y'' - 4y' + 4y = e2t given that e2t already solves the homogeneous equation.
  3. Common mistakeWhy doesn't undetermined coefficients just fail outright when the forcing term matches a homogeneous solution?
Your turn1 of 2
Find the particular solution yp(t) of y'' - (3·2)y' + 32 y = 4e3t, given that r = 3 is a repeated root (so ert already solves the homogeneous equation).

3.5 Solving a Nonhomogeneous Initial Value Problem

  1. Worked exampleSolve y'' + y = 2t with y(0) = 0 and y'(0) = 1.
  2. Worked exampleSolve y'' - y' - 2y = e3t with y(0) = 0 and y'(0) = 0.
  3. Going furtherWhat happens to the constants in the homogeneous part if the initial conditions are mistakenly applied before adding yp?
Your turn1 of 2
Solve y'' + y = 3t with y(0) = 1 and y'(0) = 2. Find y(t).

3.6 Variation of Parameters

  1. How it worksHow does letting the constants in the homogeneous solution become functions of t build a particular solution?
  2. How it worksHow does the Wronskian of the homogeneous solutions show up inside the variation of parameters formula?
  3. Worked exampleFind a particular solution of y'' + y = tan(t) using variation of parameters.
  4. Common mistakeWhy doesn't variation of parameters need the forcing term to match any particular guessable shape at all?
Your turn1 of 2
For y1 = 4cos(t) and y2 = 4sin(t), both solutions of y'' + y = 0, find their Wronskian W(t) = y1 y2' - y1' y2.

3.6 Solving with Variation of Parameters

  1. Worked exampleSolve y'' + 4y = 3 csc(2t) using variation of parameters.
  2. Going furtherWhat happens to variation of parameters when the coefficients in the differential equation aren't constant?
  3. Ties togetherHow does the variation of parameters formula connect to the first-order integrating factor method?
Your turn1 of 2
For y'' + 42 y = 4csc(4t), using variation of parameters with y1=cos(4t), y2=sin(4t), find u1'(t) = -y2(t)g(t)/W(t), where W = 4 is the Wronskian. (Notice sin and csc cancel.)

3.6 Choosing Between Undetermined Coefficients and Variation of Parameters

  1. How it worksHow do you decide whether a forcing term's shape qualifies it for undetermined coefficients?
  2. Common mistakeWhy isn't undetermined coefficients ever usable on equations with variable, non-constant coefficients?
  3. Worked exampleFind which method applies to y'' + y = sec(t) and explain why undetermined coefficients can't handle it.
  4. Ties togetherHow does variation of parameters' reliance on the Wronskian connect it back to the theory of linear independence?
Your turn1 of 2
Presented with y'' + 42 y = sec(4t), a student tries to guess a particular solution of the form A cos(4t) + B sin(4t), following the usual undetermined coefficients pattern for trigonometric forcing. Why does this fail here?