Find the solutions of the following clairaut differential equations. y=xy

alesterp

alesterp

Answered question

2021-09-18

Find the solutions of the following clairaut differential equations.
y=xy+(y)2

Answer & Explanation

Mayme

Mayme

Skilled2021-09-19Added 103 answers

Step 1
The clairaut differential equation is,
y=xy+(y)2...(1)
Differentiate the above equation with respect to x,
y'=y'+xy''+2y'(y'')
(x+2y')y''=0
Step 2
If y'' = 0,
Solving the obtained equation,
y''=0
dydx=0
dy=0dx+c
y'=c
Substitute c for y’ in equation (1).
y=xc+c2
Thus, the general solution of the clairaut differential equation is y=xc+c2.
Step 3
If x+2y'=0,
Solving the obtained equation.
x+2y'=0
y=x2
dydx=x2
dy=12xdx
y=12[x22]
y=x24
Thus, the singular solution of the clairaut differential equation is y=x24.

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