Question

solve the differential equation. y=x[p+√1+p²​

Answer 1

SOLUTION

TO SOLVE

The differential equation

$\sf{y = xp + \sqrt{1 + {p}^{2} } }$

EVALUATION

Here the given differential equation is

$\sf{y = xp + \sqrt{1 + {p}^{2} } }$

This equation is of the form

$\sf{y = xp + f(p) }$

Which is Clairaut's equation

Now for Clairaut's equation the general solution is obtained replacing p by c

Hence the required solution is

$\sf{y = cx+ \sqrt{1 + {c}^{2} } }$

Where C is constant

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