Question

Make and solve the given equation x′′ − x′ = 6 + e^{2}t

Second order linear equations
ANSWERED
asked 2020-11-08
Make and solve the given equation \(x′′ − x′ = 6 + e^{2}t\)

Answers (1)

2020-11-09
Given: \(x′′ − x′ = 6 + e^{2}t\)
Recall: The general solution to \(a(x)y′′ + b(x)y′ + c(x)y = g(x)\) is given by
\(y = y_{h} + y_{p},\)
where y_{h}, the complimentary function is the solution to the homogenens
system \(a(x)y′′ + b(x)y′ + c(x)y = 0\)
and \(y_{p}\), the particular solution, is any
function that satisfies the non-homogeneous equation.
Here it is given that \(y” - y’ = 6 + e^{2}t.\) Now for the complimentary function,
put \(y = e^{m}x\) in the homogeneous form
\(y” - y’ = 0,\) which is given by
\(m^{2} - m = 0\). Therefore
\(y_{n} = a + be^{x},\)
where a, b are arbitrary constant.
Now leeting \(D = \frac{d}{dx}\) we have
\(y_{p} =\frac{1}{D^{2}-D}(6+e^{2}t)\)
\(=\frac{1}{D^{2}-D}6+\frac{1}{D^{2}-D}e^{2}t\)
\(=6\frac{1}{(-D)}(1-D)^{-1}+e^{2}t\frac{1}{2^{2}-2}\)
\(=6\frac{1}{(-D)}(1+D+D^{2}+\cdots)+e^{2}t \frac{1}{2}\)
\(=6\frac{1}{(-D)}+\frac{e^{2}t}{2}\)
\(=-6t+\frac{e^{2}t}{2}\)
Therefore the general solution is given by
\(y = a + be^{x} + -6t + \frac{e^{2}t}{2}\)
where a, b are arbitrary constant.
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