solve differential equation for p; x²p³+yp²+p²x²y²+py³=0

p²(x²p+y) +y²p(px²+y)=0

(p²+y²p) (x²p+y)=0

p(p+y²) (x²p+y)=0

p=0

y-c=0

p+y²=0

dy/dx+y²=0

dy/y²=-dx

-1/y=-x+c

-1=-xy-cy

(cy-xy+1)

x²p+y=0

dy/dx+y/x²=0

integral factor=e^ʃ1/x²dx=e^-1/x

ye^-1/x=c

(y-c)(cy-xy+1)(ye^-1/x-c)=0

solve the differential equation for p ; xy(y-px)= x+yp

Xy²- pyx²-x-yp = 0

xy²-p(yx²+yp) –x= 0

xy²-x= p(yx²+y)

p= xy²-x/ yx²+y

dy/dx= x(y²-1)/ y(x²+1)

y/(y²-1) dy= x/ (x²+1)dx

 

let,

 y²-1= t

2ydy= dt

y dy= dt/2

and x²+1= v

2x dx= dv

x dx= dv/2

then,

dt/2t= dv/2v

1/2logt=1/2logv+logc

logt= logcv

t=  cv

(y²-1)= (x²+1)c

y²= (x²+1)c+1

solution of differential equation - (D^2 - 2D + 4)y = e^2x cos x.

The Auxiliary equation is,                                                                          

m² - 2m + 4 = 0

m = [2 ± √(4 – 16)] ∕ 2

m = 1 ± 2√3i/3

m = 1± √3i

complementary function = eˣ [A cos √3x + B sin√3x]

                                         = Aeˣ [cos √3x+B]

Particular integral=1/(D²-2D+4) e²ˣ cos x

                            = e²ˣ 1/[(D+2)²-2(D+2)+4] cos x

                            = e²ˣ 1/(D²+2D+4) cos x

                            = e²ˣ 1/(-1+2D+4) cos x

                            = e²ˣ 1/(2D+3)*(2D-3)/(2D-3) cos x

                            = e²ˣ 1/(4D²-9) 2 sin x- 3cos x

                            = -e²ˣ/13 [2 sin x- 3cos x]

y = complementary function + particular integral

y = eˣ [ A cos √3x + B sin √3x] - e²ˣ/13 [2 sin x – 3 cos x]