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)`.

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