Question

solve clairaut's equation (px+y)^2=x^2p

Answer 1

Answer:

The equation of the form y = px +f(p) where p= dy/dx is known as Clairaut’s equation named after the French mathematician Alexis Claude Clairaut (1687–1765).

The equations which are reducible to Clairaut’s form can be done so by suitable substitution.

The required substitution depends on the problem.

For example, consider x^2(y-px)=y(p^2).

This is not in Clairaut’s form.

But if we put x^2 = X and y^2= Y then p= (x/y) P where P= dY/dX.

The equation then becomes X(y- x^2P/y) = y (x^2/y2)P^2

or X(Y-XP) = X(P^2) or Y = XP + P^2 which is now in Clairaut’s form

The solution got by just replacing P by constant c.

Hence the solution of the example is Y= c X+ c^2 or y^2 = cX^2 + c^2.

Answer 2

Answer:

Step-by-step explanation:Answer:

The equation of the form y = px +f(p) where p= dy/dx is known as Clairaut’s equation named after the French mathematician Alexis Claude Clairaut (1687–1765).

The equations which are reducible to Clairaut’s form can be done so by suitable substitution.

The required substitution depends on the problem.

For example, consider x^2(y-px)=y(p^2).

This is not in Clairaut’s form.

But if we put x^2 = X and y^2= Y then p= (x/y) P where P= dY/dX.

The equation then becomes X(y- x^2P/y) = y (x^2/y2)P^2

or X(Y-XP) = X(P^2) or Y = XP + P^2 which is now in Clairaut’s form

The solution got by just replacing P by constant c.

Hence the solution of the example is Y= c X+ c^2 or y^2 = cX^2 + c^2.