Question

Prob.22. Solve the differential equation using clairaut's equation -
dy /dx-dx/dy=x/y-y/x
​

Answer 1

dy/dx-dx/dy=x/y - y/x

dy/dx-dx{1/(dy/dx) }=(x²-y²) /xy

(dy/dx)²- 1= {(x²-y²)/xy} dy/dx

(dy/dx)² - {(x²–y²) /xy} dy/dx -1=0

dy/dx=[{(x²-y²)/xy}(+Or-)√{(x²-y²)²/(xy)²+ 4}]/2

2 dy/dx={(x²-y²)/xy}(+or-)[√{(x²-y²)²-4x²y²}]/xy

2dy/dx=(x²-y²)/xy(+or-){√(x²+y²)²}/xy

2 dy/dx= (x²-y²)/xy(+or-) (x²+y²)/xy

2 dy/dx=(x²-y²)/xy + (x²+y²)/xy (using +ve value)

2dy/dx=2x²/xy

dy/dx=x/y

ydy=xdx

∫ydy=∫xdx +c₁(c₁=integrating const.)

y²/2=x²/2+c₁

y²-x²=c (c=2c₁) ……………(1)

now using -ve value

2 dy/dx =(x²-y²)xy-(x²+y²)/xy

2 dy/dx = -2 y/x

dy/y+dx/x=0

∫dy/y+∫dx/x=c₂ (c₂ =integrating const.)

ln y + ln x =ln c' ( c₂=ln c' )

ln(xy)= ln c'

xy=c ……………………………………(2)

from (1) and (2) we get the solution of given differential equation,

y²-x²= c & xy=c'

HOPE IT HELPS YOU BUDDY.