Differential Equations Answers

Solve the following differential equations.

(i) dx/y²(x-y) =dy/x²(x-y) =dz/z(x²+y²)

(ii) (yz+z²)dx - xzdy + xydz =0

(iii) (z+z³)cosx dx - (z+z³)dy + (1-z²) (y-sinx)dz =0

Show that , for the DE
d²y/dx² + a(x) dy/dx+b(x)y=0

exp(mx) is a particular integral if m²+am+b=0 . Hence find the value of m so that exp(mx) is a particular integral of the equation
(x-2) d²y/dx² - (4x-7)dy/dx+(4x-6)y=0

Reduce the equation x²(y-px)=yp² to clairaut´s form and hence find its complete solution.

Find the general solution of the DE
y² lny = x py+p².
Does the equation has any singular solution? If yes, obtain it.

y'''+y''=8x^2

Kindly assist me solving the following questions
1. (x^2y^2+xy)ydx + (x^2y^2-1)xdy=0
2. (x^2+y^2)dx + xdy=0
3. dy/dx +y=3y^3e^x

Solve the heat conduction equation:
for the following boundary and initial conditions:
8(d^2u/dt^2)=(du/dt), for 0<X< 5 and t> 0,
u(0,t) = u(5,t) = 0,
u(x,0) =2sin(πx)− 4sin(2πx)

Q. Choose the correct answer.
Q. The equation u_xx 〖-4u〗_xy=0 is
)parabolic
b)hyperbolic
c)Elliptic
d) None of above

Q. The equation u_xx 〖-4u〗_xy+ 4uyy=0 ?
)parabolic
b)hyperbolic
c)Elliptic
d) None of above

Q.Choose the correct answer.
If one of the solution of the differential equation (x-1)u’’-xu’+u=0, x>1 is
U=ex, then which of the following is the second linearly independent solution?
a)u=xex
b) u=3ex
c)u=x
d)u=x2+1