Homework Sourseaverify that the functions y1 and y2 are solutions of the sp
a.verify that the functions y1 and y2 are solutions of the specified differential equation
b. verify that y1 and y2 are linearly independent
c.find the general solution to the differential equation
d. find the solution to the initial value problem
y\'\'-2y\'+y=0 y1(x)=e^x y(2)=xe^x y(0)=1 y\'(0)=1
b. verify that y1 and y2 are linearly independent
c.find the general solution to the differential equation
d. find the solution to the initial value problem
y\'\'-2y\'+y=0 y1(x)=e^x y(2)=xe^x y(0)=1 y\'(0)=1
Solution
y1(t) = e^-3t then dy1/dt = -3e^-3t d2y1/dt2 = 9e^(-3t) Then d^2*y / dt^2 + 2*dy/dt - 3y = 9e^(-3t) + 2(-3e^-3t) - 3e^(-3t) = 9e^(-3t) - 6e^(-3t) - 3e^(-3t) = 9e^(-3t) - 9e^(-3t) = 0 So y1(t) = e^(-3t) satisfies the differential equation. For y2(t), we have y2(t) = e^t, dy2/dt = e^t, d2y2/dt2 = e^t Then d^2*y / dt^2 + 2*dy/dt - 3y = e^t + 2e^t - 3e^t = 3e^t - 3e^t = 0 Hence y2(t) = e^t satisfies the differential equation.
