Differential Equations Answers

Given that z = ax +sqrt (a^2 − 4y) + c is the complete integral of the PDE, p^2 − q^2 =4
determine its general integral.

Find the partial differential equation arising from f[z/x^3,y/z]=0, where f is an
arbitrary function of the arguments. Also find the general solution of the PDE
obtained.

Show that 2z=(ax+y)^2+b,where a,b are arbitrary constants is a complete
integral of px+qy-q^2=0

a) Verify that the total differential equation
yz dx +(x^2y-zx)dy+(x^2z-xy)dz=0 is integrable and hence find its integral.

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