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Verify the equation z(z+y^2)dx + z(z+x^2)dy - xy(x=y)=0 is intergrable and hence find its primitive

Solve the partial differential equation p cos(x+y) + q sin( x+y)=z, where the partial derivatives of z with respect to x,y are denoted by p and q respectively.

(1+yx)x dy+(1-yx)y dx=0

Reduce the following PDE to a set of three ODEs by the method of separation of

variables.

∂²u/∂r² + (2/r) ∂u/∂r + (1/r²) ∂²u/∂θ² + (cosθ/r²) ∂u/∂θ + (1/r² + sin²θ) ∂²u/∂φ² = 0

variables.

∂²u/∂r² + (2/r) ∂u/∂r + (1/r²) ∂²u/∂θ² + (cosθ/r²) ∂u/∂θ + (1/r² + sin²θ) ∂²u/∂φ² = 0

Obtain all the first and second order partial derivatives of the function:

f (x, y) = x² sin y+ y² cos x

f (x, y) = x² sin y+ y² cos x

Use the Frobenius method to obtain one solution of the following ODE:

4 yx ′′ + 2y′ + y = 0

4 yx ′′ + 2y′ + y = 0

Solve the initial value problem :

d²x/dt² -6dx/dt + 9x = 0 , x(0) =6 ,x(0) = -1 .

d²x/dt² -6dx/dt + 9x = 0 , x(0) =6 ,x(0) = -1 .

1. a) Solve the following ordinary differential equations:

i) dy/dx + 4xy = x

ii) d²y/dx² + 4dy/dx - 12y = cos2x .

i) dy/dx + 4xy = x

ii) d²y/dx² + 4dy/dx - 12y = cos2x .

Solve:(i)(dy/dx-1)^2(d^2y/dx^2+1)^2y=sin^2(x/2)+e^x+x

(ii)2x^2y(d2y/dx2)+4y^2=x^2(dy/dx)^2+2xy(dy/dx)

(ii)2x^2y(d2y/dx2)+4y^2=x^2(dy/dx)^2+2xy(dy/dx)

the normal form of the differential equation y"-xy'+(4x^2-1)y=?