Differential Equations Answers

Verify that the equations

i) z =sqrt (2x + a )+ sqrt(2y + b) and

ii)z^2+u=2(1+l ^x)(x+ly)

are both complete integrals of the PDEz=1/p+1/q .

Also show that the complete integral

(ii) is the envelope of one parameter sub-system obtained by taking b=-a/l -μ/1+l in the

solution (i)

Find the general integral of the equation

(x − y) p + (y − x − z) q = z

and a particular solution through the circle z =1, x^2 + y^2 =1

Find the differential equation of the space curve in which the two families of

surfaces

u = x^2 − y^2 = c1 and

v = y^2 − z^2 = c2 intersect.

Using the method of undermined coefficients, find the general solution of the

differential equation

y^iv-2y'''+2y''=3e^-x+2e^-xx+e^-x sin x

Identify the type of the differential equation y =xy'+1-lny' and hence

solve it

Solve the following initial value problem

d^2y/dx^2+dx/dy-2y=-6sin 2x-18cos2x

y(0)=2,y'(0)=2

Solve, using the method of variation of parameters

d^2y/dx^2-y=2/1+e^x

a) Solve the differential equation

dy/dx+(x/1-x^2)y=x (sqrt y),y(0)=1

Given that

y( x )= x^-1 is one solution of the differential equation

2x^2y''+3xy'-y=0,x>0,

find a second linearly independent solution of the equation

a) Solve the differential equation

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