5. Consider the ODE tx' + 2x = t, for t > 0. We will solve it using methods similar to those for a Cauchy-Euler ODE, and variation of parameters. (If you can think of another way of solving this ODE and want to check your answer you may of course do so, but what we want to see here is your answers to the questions below.) (a) First, we will find the homogeneous solution ch to the homogeneous ODE. Find the value of m such that xh = ctm. What can you say about the values that the constant c can take on? (b) Now, we will find a particular solution xp to the inhomogeneous ODE. To do this, assume Xp = utm, where u = function u(t) such that x, is a particular solution of the ODE. (c) Give the general solution of the ODE. No need to explain. u(t) is a function of t and m is the value you found above. Find a
5. Consider the ODE tx' + 2x = t, for t > 0. We will solve it using methods similar to those for a Cauchy-Euler ODE, and variation of parameters. (If you can think of another way of solving this ODE and want to check your answer you may of course do so, but what we want to see here is your answers to the questions below.) (a) First, we will find the homogeneous solution ch to the homogeneous ODE. Find the value of m such that xh = ctm. What can you say about the values that the constant c can take on? (b) Now, we will find a particular solution xp to the inhomogeneous ODE. To do this, assume Xp = utm, where u = function u(t) such that x, is a particular solution of the ODE. (c) Give the general solution of the ODE. No need to explain. u(t) is a function of t and m is the value you found above. Find a
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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Transcribed Image Text:5.
Consider the ODE tx' +2x = t, for t > 0. We will solve it using methods similar to those
for a Cauchy-Euler ODE, and variation of parameters. (If you can think of another way of solving
this ODE and want to check your answer you may of course do so, but what we want to see here is
your answers to the questions below.)
(a) First, we will find the homogeneous solution xh to the homogeneous ODE. Find the value of
= ctm. What can you say about the values that the constant c can take on?
(b) Now, we will find a particular solution xp to the inhomogeneous ODE. To do this, assume
u(t) is a function of t and m is the value you found above. Find a
m such that xh
utm, where u
Xp
function u(t) such that x, is a particular solution of the ODE.
(c) Give the general solution of the ODE. No need to explain.
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