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.

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.