Solution Found!
Discontinuous Forcing Term. In certain physical models,
Chapter 4, Problem 41E(choose chapter or problem)
Discontinuous Forcing Term. In certain physical models, the nonhomogeneous term, or forcing term, \(g(t)\) in the equation
\(a y^{\prime \prime}+b y^{\prime}+c y=g(t)\)
may not be continuous but may have a jump discontinuity. If this occurs, we can still obtain a reasonable solution using the following procedure. Consider the initial value problem
\(y^{\prime \prime}+2 y^{\prime}+5 y=g t\) ; \(y(0)=0\) , \(y^{\prime}(0)=0\) ,
where
\(g(t)= \begin{cases}10 & \text { if } 0 \leq t \leq 3 \pi / 2 \\ 0 & \text { if } t>3 \pi / 2\end{cases}\).
(a) Find a solution to the initial value problem for \(0 \leq t \leq 3 \pi / 2\)
(b) Find a general solution for \(t>3 \pi / 2\)
(c) Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first derivatives, at \(t=3 \pi / 2\). This gives us a continuously differentiable function that satisfies the differential equation except at \(t=3 \pi / 2\).
Equation Transcription:
Text Transcription:
g(t)
ay''+by'+cy=g(t)
y''+2y'+5y=gt
y(0)=0
y'(0)=0
g(t)={_0 if t>3 pi/2 ^10 if 0 t</= 3pi/2
0 </=t </= 3 pi/2
t> 3 pi/2
t=3 pi/2
t=3 pi/2
Questions & Answers
QUESTION:
Discontinuous Forcing Term. In certain physical models, the nonhomogeneous term, or forcing term, \(g(t)\) in the equation
\(a y^{\prime \prime}+b y^{\prime}+c y=g(t)\)
may not be continuous but may have a jump discontinuity. If this occurs, we can still obtain a reasonable solution using the following procedure. Consider the initial value problem
\(y^{\prime \prime}+2 y^{\prime}+5 y=g t\) ; \(y(0)=0\) , \(y^{\prime}(0)=0\) ,
where
\(g(t)= \begin{cases}10 & \text { if } 0 \leq t \leq 3 \pi / 2 \\ 0 & \text { if } t>3 \pi / 2\end{cases}\).
(a) Find a solution to the initial value problem for \(0 \leq t \leq 3 \pi / 2\)
(b) Find a general solution for \(t>3 \pi / 2\)
(c) Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first derivatives, at \(t=3 \pi / 2\). This gives us a continuously differentiable function that satisfies the differential equation except at \(t=3 \pi / 2\).
Equation Transcription:
Text Transcription:
g(t)
ay''+by'+cy=g(t)
y''+2y'+5y=gt
y(0)=0
y'(0)=0
g(t)={_0 if t>3 pi/2 ^10 if 0 t</= 3pi/2
0 </=t </= 3 pi/2
t> 3 pi/2
t=3 pi/2
t=3 pi/2
ANSWER:
Solution
Step 1 of 11
In this problem we have to determine the solution for the differential equation which has a non homogeneous discontinuous term.
Given differential equation
where
if
if