Question

Solve the given Clairaut's equation
x²(y-px) = p²y​

Answer 1

Answer:

Let's put X=x^2;\quad Y=y^2X=x

2

;Y=y

2

then p=\frac{x}{y}\frac{dY}{dX}p=

y

x

dX

dY

. Let's denote P=\frac{dY}{dX}P=

dX

dY

Then the equation can be rewritten in form

X(y-x^2\frac{P}{y})=y\frac{x^2}{y^2}P^2X(y−x

2

y

P

)=y

y

2

x

2

P

2

By multiplying both sides with y, we assume

X(y^2-x^2P)=x^2P^2X(y

2

−x

2

P)=x

2

P

2

Or

X(Y-XP)=XP^2X(Y−XP)=XP

2

Therefore

Y=XP+P^2Y=XP+P

2

which is now in Clairaut’s form

The solution got by just replacing P by constant c.

Hence

Y=cX+c^2Y=cX+c

2

or

y^2=cx^2+c^2y

2

=cx

2

+c

2

I hope it will help you dear ❣️❤️☺️