Differential Equations with Boundary Value Problems HigHer-Order differential equation and Modeling with Higher-order differential equation 1. Solve: x?y”

Differential Equations with Boundary Value Problems HigHer-Order differential equation and Modeling with Higher-order differential equation 1. Solve: x?y” + xy’ +1²y = 0 with the Boundary Conditions: y(1) = 1 and y(ve) = 1
2. Solve: xy” + 2y’ – 4xy = 0 about the point x, = 0
3. The point Xo = 0 is a regular point for the following ODE. Use power series to find the solution
around Xo = 0. You need to find only the first three terms of the series.
y’ +sin(x) y = 0, y(0) = 6
4. Find the solution about the point x = 1 to the following initial value problem.
xy” + 3(x – 1)y’ + 6y = 0, y(1) = 4, y'(1) = 9
Please show terms out to the fifth power.
5. A spring with spring constant 12 pounds per foot is attached to the ceiling. A 4 pound weight is
attached and it is set in motion by the mass is pushed up three inches from the equilibrium position and
given a downward velocity of 2 feet per second. The system is in a medium that imparts a damping force
numerically equal to 2 times the velocity. Solve this IVP.
6. Prove that if yp, is a particular solution of Po(x)y” +P+(x)y’ + P2(x)y = F1(x) for x € (a,b)
and Yp, is a particular solution of Po(x)y” +P_(x)y’ + P2(x)y = F2(x) for x € (a, b)
then yp = yp. + yp, is a solution of Po(x)}” + P2(x)y’ + P2(x)y = F2(x) + F2(x) for x € (a,b).
7. Show that if az(x)y” + a1(x)y’ + a.(x)y = f(x) then
g(x)yz dx + y2S
y = 61+ c2yz – y?s
g(x)y?
W(172)
W(7172)
dx, where g(x) = f(x)
az(x)

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