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Question

# Third-Order Examples: For each of the nonhomogeneous linear DEs in $(a)$ Ver(fy that the given $y_{1}, y_{2}, y_{3}$ satisfy the corresponding homogeneous equation. $(b)$Use the Superposition Principle, with appropriate coefficients, to state the general solution $y_{1}(t)$ to the corresponding homogeneous equation. $(c)$ Verify that the given $y_{p}(t)$ is a particular solution to the given nonhomogeneous s DE. $(d)$ Use the Nonhomogeneous Principle to write the general solution $y(t)$ to the nonhomogeneous DE. $(e)$ Solve the NP consisting of the nonhomogeneous DE and the given initial conditions$\begin{array}{l}{y^{\prime \prime \prime}+y^{\prime \prime}-y^{\prime}-y=4 \sin t+3} \\ {y_{1}=e^{t}, y_{2}=e^{-t} \cdot y_{3}=t e^{-t}} \\ {y_{p}=\cos t-\sin t-3} \\ {y(0)=1 . y^{\prime}(0)=2, \quad y^{\prime \prime}(0)=3}\end{array}$

Solution

Verified
Step 1
1 of 7

The corresponding homogeneous equation is $y''' - y'' +y' +y = 0$

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