Third Midterm—Practice Test

In the exam, we will be using Table 6.2.1 from Boyce-DiPrima as aid in our problems involving Laplace transforms. Feel free to use that table in these problems as well.

Show that $y_1=x^3$ and $y_2 = \lvert x^3 \rvert$ are linearly independent solutions on the real line of the equation $xy'' - 3xy' +3y =0.$ Verify that the Wronskian $W(y1,y2)$ is identically zero. Why do these facts not contradict Theorem 3 in page 154?
A second-order Euler equation is one of the form
$ax^2 y''+bxy' +cy = 0$

where $a,b,c$ are constants. Show that for non-negative values of $x$ the substitution $v=\ln x$ transforms the previous equation into the constant-coefficient linear equation

$a \dfrac{d^2y}{dv^2} + (b-a) \dfrac{dy}{dv} + cy =0$

with independent variable $v.$ If the roots $r_1$ and $r_2$ of the characteristic equation of the previous differential equation are real and distinct, conclude that a general solution of the Euler equation is $y=c_1 x^{r_1} + c_2 x^{r_2}.$
Find a function $y(x)$ such that $y^{(4)}(x)=y^{(3)}(x)$ for all $x,$ and $y(0)=18, y'(0)=12, y''(0)=13,$ and $y^{(3)}(0)=7.$
Find a particular solution of the equation $2y''+4y'+7y=x^2$ using both the method of variation of parameters, and undetermined coefficients.
Find the solution of the equation $y^{(4)}-y^{(3)}-y''-y'-2y=8x^5$ that satisfies the initial conditions $y(0)=y'(0)=y''(0)=y^{(3)}(0)=0.$ Use both undetermined coefficients and methods based on the Laplace transform.
Find the Laplace transform of the function $f(x) = \sin 3x \cos 3x.$
Find the inverse Laplace transform of the function $F(s)= \dfrac{1}{(s^2+b^2)^2}.$
Find the inverse Laplace transform of the function $F(s) = \dfrac{s^2+1}{s^3-2s^2-8s}.$
Solve the initial value problem $y^{(4)}+2y''+y=4xe^x$ with $y(0)=y'(0)=y''(0)=y^{(3)}(0)=0$ using exclusively methods based on the Laplace transform.
Find the Laplace transform of the functions $f(x)=\dfrac{\sin x}{x}$ and $g(x)=xe^{2x}\cos 3x.$