Question

y = 2px +y²p³
solve using clairaut equation​

Answer 1

Answer:

The given equation is written as

x=(1/2) ( (y/p) -y² p²

Differentiating w. r. t. y, we get

Answer 2

Answer:

 $y^{2}$= xc +   $\frac{c^{3} }{8}$  is the solution.

Step-by-step explanation:

Clairaut's equation is a differential equation of the form

$y = x\frac{dy}{dx}+f( \frac{dy}{dx})$

To solve Clairaut equation, the following steps are followed:

1. Differentiate the given equation with x.
2. Separate it into product of two terms and equate them to zero.
3. Substitute the result into the equation.
4. This gives the general solution of Clairaut's equation.
5. The general solution is of the form y = Cx + f(C).
6. The other term gives the singular solution.

Given equation is y = 2px +y²p³

Step 1: Reduce the equation to clariut equation form

Multiply both sides of equation with y.

$y^{2} = 2pxy + y^{3} p^{3}$

Put $y^{2}$ = t

$2y\frac{dy}{dx} = \frac{dt}{dx}$

2yp = p

t = xp + $\frac{p^{3} }{8}$

This is in the form of clairaut's equation.

The solution is p = c

$y^{2}$ = xc + $\frac{c^{3} }{8}$

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