Differential Equations Answers

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.

Solve the initial value problem: d^2x/dt^2 - 6 dx/dt +9x =0, x(0) =6, X^. (0) =-1

Solve the following ordinary differential equations: i) dy/dx + 4xy = x, (2) d^2y/dx^2 + 4 dy/dx - 12 y =cos 2x

Use the power series method to obtain one solution of the following ODE:x^2y"+3y'-xy=0

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

Solve the DE : {(dy/dx - 1)^2)} ×{ (d^2y/dx^2+1)^2} ×y =, sin^2(x/2)+e^x+x.

Solve x×dy/dx +y ln y =x×y×e^x.