Differential Equations Answers

The differential equation of a damped vibrating system under the action of an external periodic force is: d^2x/dt^2 +2 m0 dx/dt +n^2x = a cos pt Show that, if n>m0>0 the complementary function of the differential equation represents vibrations which are soon damped out. Find the particular integral in terms of periodic functions

Reduce the following PDE to a set of three ODEs by the method of separation of variables. d^2u/dr^2 +(2/r)(du/dr)+(1/r^2)(d^2u/dθ ^2) +(cot θ/r^2)(du/dθ) + (1/r^2 sin^2 θ)(d^2u/d φ^2) =0

Solve the given differential equation by using appropriate substitution.

dy/dx = (1-x-y) / (x+y)

solve the given initial value problem. Give the largest interval I over which the solution is defined.

(1+t^2)dy/dx + x = tan^-1 t , x(0) = 4

[Hint: In your solution let u = tan^-1 t ]

Solve the given differential equation by separation of variables.

(e^x + e^-x)dy/dx=y^2

determine a region of the xy-plane for which the given DE would have a unique solution whose graph passes through a point (x0,y0) in the region

(y-x)y'=y+x

Verify that the indicated expression is an implicit solution of the given first order differential equation.Find at least one explicit solution y=Φ(x) in each case. Use a graphing utility to obtain the graph of an explicit solution.
Give an interval I of definition of each solution Φ.

2xydx+(x^2-y)dy=0 , -2x^2 y + y^2 =1

determine a region of the xy-plane for which the given DE would have a unique solution whose graph passes through a point (x0,y0) in the region

(y-x)y'=y x

Solve the initial value problem

(e^x + y)dx + (2 + x +ye^y)dy=0, y(0)=1

Show that the function i) u(x, t) =A(x+ct) ^3 is a solution of the one-dimensional wave equation. ii) u(x, t) = {(e)^-(mu ×t)} sinx is a solution of the one-dimensional heat equation.