91Ó°ÊÓ

Verify that the given function is in the kernel of \(L\). $$y(x)=x e^{2 x}, \quad L=D^{2}-4 D+4$$

Determine the annihilator of the given function. $$F(x)=4 e^{-2 x} \sin x$$.

For problems \(5-7\) you will need to use the function inner product \(=\int_{a}^{b} f(x) g(x) d x\) in \(C^{2}[a, b].\) Determine a basis for the solution space to \(y^{\prime \prime}+y^{\prime}-\) \(2 y=0\) that is orthogonal on the interval [0,1].

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}-4 y^{\prime}+5 y=e^{2 x} \tan x, \quad 0

Find \(L y\) for the given differential operator \(L\) and the given function \(y.\) $$L=4 x^{2 D}, \quad y(x)=\sin ^{2}\left(x^{2}+1\right).$$

Determine the general solution to the given differential equation on \((0, \infty)\) $$x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0$$

For problems \(5-7\) you will need to use the function inner product \(=\int_{a}^{b} f(x) g(x) d x\) in \(C^{2}[a, b].\) Determine a basis for the solution space to \(y^{\prime \prime}+4 y=0\) that is orthogonal on the interval \([0, \pi / 4].\)

Show that the differential equation governing the behavior of an RLC circuit can be written directly in terms of the current \(i(t)\) has $$ \frac{d^{2} i}{d t^{2}}+\frac{R}{L} \frac{d i}{d t}+\frac{1}{L C} i=\frac{1}{L} \frac{d E}{d t} $$

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation. $$y^{\prime \prime}+y=3 e^{x} \cos 2 x$$

Determine the annihilator of the given function. $$F(x)=e^{x} \sin 2 x+3 \cos 2 x$$.