Additional Exercises for Chapter 2

Analytical Methods for Differential Equations

Notation: \(y' = dy/dx, \ \dot x = dx/dt \).
  1. Find the general solutions of the following first-order differential equations:
    1. $$x y'= 3y$$
    2. $$y' + 2 xy = x^3$$
    3. $$\dot x= 2xt+4t+3x+6$$
    4. $$\dot x + x= e^t$$
    5. $$x y' + 2 y = x^3 y^2$$
  2. Find the specific solutions of the following first-order differential equations that obey the given conditions:
    1. $$y'=-x^2 y^4, \qquad y(0)=1$$
    2. $$y'-y\tan x = \sin x, \qquad y(0)=1$$
    3. $$x y'-\frac{y}{1+x}=6x^3, \qquad y(1)=3$$
    4. $$\dot x =t+tx^2, \qquad x(1)=0$$
  3. Suppose that \(y' = f(y/x) \) for some function \( f\). Show that if \( u(x) = y(x)/x\), then \(u \) obeys a separable differential equation. Hence find the general solution of $$ y' = \frac{ x^2+y^2}{xy}. $$ What is the specific solution that obeys the initial condition \(y(1)=1\)?
  4. Find the general solutions of these second-order differential equations:
    1. $$ 6 \ddot x +5\dot x-6x=0$$
    2. $$ 6 \ddot x +5\dot x-6x= 8e^{-2t} + 12$$
    3. $$ x^2 y'' + x y'-4y=0$$
    4. $$x^2 y'' + x y'-4y=9x$$
  5. Solve the following differential equations, subject to the given boundary/initial conditions:
    1. $$\ddot x-2\dot x-15x=0, \qquad x(1)=1, x\to 0 \mbox{ as } t \to\infty$$
    2. $$y''-6y'+9y=0, \qquad y(0)=0, y(1)=1$$
    3. $$ y'' + 16y = \cos( 3x), \qquad y(0)=1, y'(0)=0$$
  6. Find the general solution of the third-order equation $$y'''+y''+y'+y=0. $$ Then find the specific solution for which \(y\) is bounded, \(y(0)=0\) and \(y'(0)=1\). How do the general and specific solutions change if the RHS of the equation is changed from 0 to \( \sin(2x)\)?

Answers

Home