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.

  1. 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?
  2. 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}.

  3. 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.
  4. Find a particular solution of the equation 2y''+4y'+7y=x^2 using both the method of variation of parameters, and undetermined coefficients.
  5. 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.
  6. Find the Laplace transform of the function f(x) = \sin 3x \cos 3x.
  7. Find the inverse Laplace transform of the function F(s)= \dfrac{1}{(s^2+b^2)^2}.
  8. Find the inverse Laplace transform of the function F(s) = \dfrac{s^2+1}{s^3-2s^2-8s}.
  9. 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.
  10. Find the Laplace transform of the functions f(x)=\dfrac{\sin x}{x} and g(x)=xe^{2x}\cos 3x.
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