# Questions in Section Differential Equations: page 6

Find the general solution of the differential equation (d^2y)/(dx^2)+(dy)/(dx)-2y=2sinx.

Find the solution to the following differential equation subject to the given initial conditions:

y''-3y'-4y=sin x, y(0)=0, y'(0)=1.

Consider the infinite series f(x)=Asum_(n=0)^oo((-1)^n3^(2n)x^(2n))/((2n)!) where A is a constant.

1. Show that this series ia a solution of the differential equation f''(x)+9f(x)=0.
2. Find the general solution for the differential equation f''(x)+9f(x)=0.
3. To which part of the general solution (as found in part b)) does the infinite series (above) correspond?

Show that the coefficients of dx and dy are homogeneous functions of the same degree. Then solve by making an appropriate substitution.

ydx+x(lnx-lny-1)dy=0 subjected to y(1)=e.

Show that the following differential equation is exact and then solve it.

(3xcos3x+sin3x-3)dx+(2y+5)dy=0

Solve the following linear differential equation x^2y'+x(x+2)y=e^x.

Use the Convolution Theorem to find the inverse Laplace transform of the following function: F(s)=(11s)/(s^2+121)^2.

Find the general solution to the following differential equation: (d^2y)/(dx^2)-2(dy)/(dx)-35y=1+13sin(x)-e^(3x).

Use Euler's numerical method to find the terms y_0, y_1, y_2, y_3 to the differential equation: y'=t+t^2y, y(0)=1, h=1.

Consider the following first-order nonlinear differential equation (dy)/(dx)+a(x)y^2+b(x)y+c(x)=0.

Differential equations of this form are called Riccati differential equations, and in general they are very difficult to solve. One potential solution strategy is to replace the dependent variable y(x) with a new dependent variable w(x), where y(x)=1/(a(x))xx1/(w(x))xx(dw)/(dx).