Problem 4 (Non-homogeneous Equations Redux). By now we are very familiar with a procedure for solving a linear, second-order initial value problem\[y'' + p(t)y' + q(t)y = f(t), \quad y(t_0) = y_0, \quad y'(t_0) = v_0.\]1. First, find the general solution to the associated homogeneous equation\[y'' + p(t)y' + q(t)y = 0.\]2. Find a particular solution to the non-homogeneous equation\[y'' + p(t)y' + q(t)y = f(t)\]and add it to the homogeneous solution.3. Plug in the initial conditions to find the solution to the IVP.There is actually another way we can proceed, and it allows us to plug the initial conditions into the homogeneous solution before finding the particular solution.(a) Suppose $ u_1(t) $ is a solution to the initial value problem\[y'' + p(t)y' + q(t)y = 0, \quad y(t_0) = y_0, \quad y'(t_0) = v_0,\]and $ u_2(t) $ is a solution to\[y'' + p(t)y' + q(t)y = f(t), \quad y(t_0) = 0, \quad y'(t_0) = 0.\]Show that $ y(t) = u_1(t) + u_2(t) $ is a solution to the original initial value problem (1).(b) Thanks to Variation of Parameters, we know that if $\{y_1, y_2\}$ is a fundamental set of solutions to the associated homogeneous equation, then\[y_p(t) = v_1(t)y_1(t) + v_2(t)y_2(t)\]is a particular solution, where\[v_1(t) = - \int_{t_0}^{t} \frac{y_2(\tau)f(\tau)}{y_1(\tau)y_2'(\tau) - y_1'(\tau)y_2(\tau)} \, d\tau,\quad v_2(t) = \int_{t_0}^{t}

NOTE: According to guideline answer of first question can be given, for other please ask in a…

Suppose u1(t) is a solution to the initial value problem y" + p(t)y' + q(t)y = 0, y(to) = Yo, y'(to) = vo, %3D %3D and u2(t) is a solution to y/" + p(t)y' + q(t)y = f(t), y(to) = 0, y (to) = 0. (2) %3D %3D %3D Show that y(t) = u1(t) + u2(t) is a solution to the original initial value problem (1). %3D

Suppose u1(t) is a solution to the initial value problem y" + p(t)y' + q(t)y = 0, y(to) = Yo, y'(to) = vo, %3D %3D and u2(t) is a solution to y/" + p(t)y' + q(t)y = f(t), y(to) = 0, y (to) = 0. (2) %3D %3D %3D Show that y(t) = u1(t) + u2(t) is a solution to the original initial value problem (1). %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Problem 4 (Non-homogeneous Equations Redux). By now we are very familiar with a procedure for solving a linear, second-order initial value problem

\[
y'' + p(t)y' + q(t)y = f(t), \quad y(t_0) = y_0, \quad y'(t_0) = v_0.
\]

1. First, find the general solution to the associated homogeneous equation

\[
y'' + p(t)y' + q(t)y = 0.
\]

2. Find a particular solution to the non-homogeneous equation

\[
y'' + p(t)y' + q(t)y = f(t)
\]

and add it to the homogeneous solution.

3. Plug in the initial conditions to find the solution to the IVP.

There is actually another way we can proceed, and it allows us to plug the initial conditions into the homogeneous solution before finding the particular solution.

(a) Suppose $ u_1(t) $ is a solution to the initial value problem

\[
y'' + p(t)y' + q(t)y = 0, \quad y(t_0) = y_0, \quad y'(t_0) = v_0,
\]

and $ u_2(t) $ is a solution to

\[
y'' + p(t)y' + q(t)y = f(t), \quad y(t_0) = 0, \quad y'(t_0) = 0.
\]

Show that $ y(t) = u_1(t) + u_2(t) $ is a solution to the original initial value problem (1).

(b) Thanks to Variation of Parameters, we know that if $\{y_1, y_2\}$ is a fundamental set of solutions to the associated homogeneous equation, then

\[
y_p(t) = v_1(t)y_1(t) + v_2(t)y_2(t)
\]

is a particular solution, where

\[
v_1(t) = - \int_{t_0}^{t} \frac{y_2(\tau)f(\tau)}{y_1(\tau)y_2'(\tau) - y_1'(\tau)y_2(\tau)} \, d\tau,
\quad v_2(t) = \int_{t_0}^{t}

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