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Chapter 2: Problem 5

Consider the nonhomogeneous differential equation \(x^{\prime \prime}-3 x^{\prime}+2 x=6 e^{3 t}\). a. Find the general solution of the homogenous equation. b. Find a particular solution using the Method of Undetermined Coefficients by guessing \(x_{p}(t)=A e^{3 t}\). c. Use your answers in the previous parts to write the general solution for this problem.

Short Answer

Expert verified
The general solution of the differential equation is \(x(t) = c_1e^{2t} + c_2e^{t} + 3e^{3t}\).

Step by step solution

01

Solve the homogeneous equation

The homogeneous equation is obtained by setting the right hand side of the given equation to zero. Hence the homogeneous equation we will solve is \(x'' - 3x' + 2x = 0\). The characteristic equation of this homogeneous differential equation is \(m^2 - 3m + 2 = 0\). Factoring this equation gives us \((m-2)(m-1) = 0\). Hence, the roots are \(m = 2, 1\). This means the general solution to the homogeneous equation is \(x_h(t) = c_1e^{2t} + c_2e^{t}\), where \(c_1\) and \(c_2\) are constants to be determined by the initial conditions.
02

Guess and verify a particular solution

We now find a particular solution to the non-homogeneous equation. For the non-homogeneous term in the differential equation, which takes the form \(6e^{3t}\), we can guess a particular solution of the form \(x_p(t) = Ae^{3t}\). Plug \(x_p(t)\) back into the original differential equation and solve for the constant \(A\). After calculation, we get \(A = \frac{6}{2}=3\). So a particular solution to the non-homogeneous equation is \(x_p(t) = 3e^{3t}\).
03

Write the general solution for the problem

The general solution of a nonhomogeneous differential equation is a combination of the homogeneous solution and the particular solution. Hence the general solution is \(x(t) = x_h(t) + x_p(t) = c_1e^{2t} + c_2e^{t} + 3e^{3t}\).

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Most popular questions from this chapter

Consider the system $$ \begin{array}{r} x^{\prime}=-4 x-y \\ y^{\prime}=x-2 y \end{array} $$ a. Determine the second-order differential equation satisfied by \(x(t)\). b. Solve the differential equation for \(x(t)\). c. Using this solution, find \(y(t)\). d. Verify your solutions for \(x(t)\) and \(y(t)\). e. Find a particular solution to the system given the initial conditions \(x(0)=1\) and \(y(0)=0\)

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