1 6 points each For the differential equation x2y 7xy 15y

1. (6 points each) For the differential equation x2y\'\' - 7xy? + 15y = 0. x > 0 a) Verify that y1 = x^3 and y2 = x^5 are solutions to the differential equation b) Determine whether the functions are linearly independent on x > 0 c) Form the general solution to the differential equation. 2. (10 points) Given that y1 = x^4 is a solution to the differential equation x^2 y\'\' - 7xy? + 16y = 0, x > 0 Use reduction of order method to find a second solution y2 that is linearly independent of y1 on the interval x > 0 3. (8 points each) Find the general solution to the differential equations a) y\'\'+4y?+3y = 0 b) y\'\'+10y?+25y= 0 4. (10 points) Solve the initial value problem y\'\' + 4y = 0 , y(0) = 1, y?(0) = 0

Solution

3. a) the auxiliary equation is of the form:

        D² + 4D + 3 = 0

        (D+3)(D+1) = 0

         D = -3 , D = -1

The general solution is given by:

y = C₁ e^(-3x) + C₂ e^(-x)

b) the auxiliary equation is of the form:

        D² + 10D + 25 = 0

        (D+5)(D+5) = 0

         D = -5 , D = -5

The general solution is given by:

y = C₁ e^(-5x) + C₂ xe^(-5x)