AMATH 351 Homework 5
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AMATH 351 Homework 5: Cauchy-Euler equation, nth order linear ODEs with constant coefficients, method of undetermined coefficients
Exercise 1: Roots of the characteristic equation. The characteristic equation of a homogeneous linear ODE with constant coefficients can be written as
(r − 2)3 (r2 − 6r + 25)2 (r − 3)2 = 0
1. Determine the roots of the characteristic equation and their multiplicity.
2. Find the general solution of the ODE.
Exercise 2: nth order homogeneous ODEs.
1. Find the general solution of the ODE:
y\\\ − 3y\\ + 4y\ − 2y = 0
Hint: One solution of the characteristic equation is r1 = 1. (since the sum of equation’s coefficients is zero).
2. Find the solution of the IVP
y\\\ − 5y\\ + 100y\ − 500y = 0, y(0) = 0, y\ (0) = 10, y\\ (0) = 250, if we know that e5t is one of the fundamental solutions of the ODE.
Exercise 3: Non-homogeneous Cauchy-Euler equation. Consider the
t2 
+ t 
+ y(t) = t, t > 0. (1)
1. By performing change of variable x = ln(t), see that ODE (1) is expressed equivalently as
dx2
2. Determine the general solution of the homogeneous equation that corre- sponds to ODE (2).
3. Find a particular solution of ODE (2) via the method of undetermined coefficients.
4. Determine the general solution y(北) of ODE (2). Then, use 北 = ln(t) to return to the variable t, in order to determine the general solution y(t) of the original ODE (1).
Exercise 4: Undetermined coefficients and superposition principle.
Using the method of undetermined coefficients, find one particular solution for each of the following non-homogeneous ODEs
1. y\\ − 3y\ + 2y = 2e2t
2. y\\ − 3y\ + 2y = 2sin(t)
By using the particular solutions you found in the previous questions, and by invoking the superposition principle, find one particular solution of the ODE:
3. y\\ − 3y\ + 2y = 2e2t + 2sin(t)
2023-05-08
Cauchy-Euler equation, nth order linear ODEs with constant coefficients, method of undetermined coefficients