Solution Exam 2 (MATH246, Spring 2024)

Problem 1

Characteristic equation has solutions .

Particular solution for is

Particular solution for is (since is root of multiplicity 2)

Particular solution for is

clearvars

syms y(t)

syms C1 C2

for f = [-t,-t*exp(-t),-t*exp(-t)*sin(t)]

f

sol = dsolve(diff(y,2)+2*diff(y)+y==f);

particular_sol = expand( subs(sol,{C1,C2},{0,0}) )

end

f = 

particular_sol = 

f = 

particular_sol = 

f = 

particular_sol = 

Problem 2

Characteristic equation has solutions . Hence and Wronskian

Particular solution is where

, hence

, hence

clearvars

syms y(t)

syms C1 C2

sol = dsolve(diff(y,2)-4*y==1+exp(2*t));

particular_sol = expand( subs(sol,{C1,C2},{0,0}) )

particular_sol = 

Problem 3

We have the ODE or .

Problem 3(a)

The natural frequency is , the external frequency is . We want , i.e. , so .

The general solution is

(here are determined using the method of undetermined coefficients).

As t goes to infinity the particular solution will dominate, giving oscillations with increasing amplitude.

Problem 3(b)

We have the ODE with characteristic equation . For critical damping we have a double root, i.e., , i.e., or . Then we have with roots .

The general solution is

(here are determined using the method of undetermined coefficients).

As t goes to infinity the terms will decay exponentially, so the solution will be very close to the particular solution .

Problem 4(a)

, hence

, .

We have , i.e.

, hence with from above.

Problem 4(b)

We have

Let , multiply by :

plugging in gives , so .

plugging in gives , so .

plugging in gives , so

This gives

syms s t

F = 1/(s*(s^2+2*s+5));

f = ilaplace(F)

f =