Method of Undetermined Coefficients Problem

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jjm5119

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Use the method of undetermined coefficients to find the solution of the equation

y''+4y = { 4t when 0<= t < pi/2 and 2pi*e^(pi/(2-t)) when t>=pi/2}

that satisfies the initial conditions y(0)=0 and y'(0)=1. Assume that y and y' are continuous functions of t. Hint: First solve the initial value problem for t<pi/2, and then solve for t>=pi/2 and then use continuity to match the solutions at t=pi/2.

Alright so I actually have no idea how to start this. I don't understand what the hint is trying to hint at. How do I go about solving for t<pi/2 or t>pi/2? Any help is appreciated.
 
The function \(\displaystyle y = t, 0 \le t \le \pi/2\) is certainly a solution by inspection. However, it isn't clear what the right hand side is supposed to be for \(\displaystyle t \ge \pi/2\). Note that for continuity, you need the right hand side to reduce to \(\displaystyle \pi/2\).
 
jjm5119 said:
Use the method of undetermined coefficients to find the solution of the equation

y''+4y = { 4t when 0<= t < pi/2 and 2pi*e^(pi/(2-t)) when t>=pi/2}

that satisfies the initial conditions y(0)=0 and y'(0)=1. Assume that y and y' are continuous functions of t. Hint: First solve the initial value problem for t<pi/2, and then solve for t>=pi/2 and then use continuity to match the solutions at t=pi/2.

Alright so I actually have no idea how to start this. I don't understand what the hint is trying to hint at. How do I go about solving for t<pi/2 or t>pi/2? Any help is appreciated.
Click to expand...
The forcing function( the right-hand side of the ODE) for t>=pi/2

Is it

\(\displaystyle e^{(\frac{\pi}{2}\,-t)}\)

or

\(\displaystyle e^\frac{\pi}{(2-t)}\)
 
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