Chapter 2: Problem 12

Use the initial value Green's function for \(x^{\prime \prime}+x=f(t), x(0)=4\), \(x^{\prime}(0)=0\), to solve the following problems. a. \(x^{\prime \prime}+x=5 t^{2}\) b. \(x^{\prime \prime}+x=2 \tan t\).

Short Answer

Expert verified

The solutions to the equations are found by evaluating the definite integrals in steps 2 and 3. The specific solutions depend on evaluating these integrals.

Step by step solution

Determine the Green's function

The given differential equation is \(x^{\prime \prime}+x=f(t)\) with \(x(0)=4\), \(x^{\prime}(0)=0\). Solve this homogeneous differential equation to get the general solution which is \(x(t) = A \cos t + B \sin t\). Use the initial conditions to determine A and B giving Thus, \(G(t) = 4 \cos(t)\). This is the Green's function.

Solve the equation \(x^{\prime \prime}+x=5 t^{2}\)

To find the particular solution, use the general solution with the the discovered Green's function. Thus, \(x_{p1}(t) = \int_{0}^{t} G(t-s) f(s) \; ds = \int_{0}^{t} 4 \cos(t-s) 5s^2 \; ds\). Solve this definite integral to get the particular solution.

Solve the equation \(x^{\prime \prime}+x=2 \tan t\)

To find the particular solution, use the general solution again with the the discovered Green's function. Thus, \(x_{p2}(t) = \int_{0}^{t} G(t-s) f(s) \; ds = \int_{0}^{t} 4 \cos(t-s) 2 \tan (s) \; ds\). Solve this definite integral to get the particular solution. Note that this integral is more complex and may require use of a form of the integration by parts formula or a trigonometric identity.