Differential Equations Answers

Find the deflection of the fixed end vibrating string of unit length corresponding to zero initial deflection and u(x) given below as the initial velocity.

x , 0 ≤ x < 1/2
u(x) =
1-x , 1/2 ≤ x <1

Find the deflection of the fixed end vibrating string of unit length corresponding to zero initial deflection and u(x) given below as the initial velocity.

x , 0 ≤ x < 1/2
u(x) =
1-x , 1/2 ≤ x < 1

Solve the following boundary value problem .
uₙ = uₓₓ , 0<x<1 , n > 0
u(0,n) = u(l, n) = 0
u(x,0) = x (l - x) , 0 ≤ x ≤1

Using Jacobi's method , Find the complete integral of the equation
2u₁xz + 3u₂y² + u₂²u₃ =0.

Find the complete integral of
p²+q²-2px-2qy+1 =0

Find the integral surface of the linear PDE
x(y²+z)p - y(x²+z)q = (x²-y²)z , which contains the straight line x+y =0 , z=1.

Find the differential equation of the family of surfaces ∅(x+y+z , x²+y²-z²) =0.
what is the order of this PDE ?

Verify that the equation 2(y+z)dx - (x+z)dy+ (2y-x+z)dz = 0 is integrable and find its primitive .

Solve:
(1+x)² d²y/dx² + (1+x) dy/dx + y = 4 cos ln(1+x)

Solve the IVP.
x² dy/dx = cosx - 2xy , y(π) = 0, x>0.