The highest derivative is of order 1 and its power is 3
∴ order is 1 and degree is 3.

The highest derivative if of order 3 and its power is 1
∴ order is 3 and degree is 1.

Squaring both sides we get,
\(\left[ \sqrt { 1+\left( \frac { dy }{ dx } \right) ^{ 2 } } \right] ^{ 2 }\)= (4x)² ⇒ 1+\(\left( \frac { dy }{ dx } \right) ^{ 2 }\)= 16x²
The highest derivative is of order 1and its power is 2
∴ order is 1 and degree is 2.

Taking power 3 both sides, we get
\(\left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 2 }{ 3 } \times 3 }=\left( \frac { d^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 3 }\)
⇒ \(\left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ 2 }=\left( \frac { d^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 3 }\)
The highest derivative is of order 2 and its power is 3
∴ order is 2 and degree is 3.

Given y = mx + \(\frac { a }{ m } \) ...(1)
Differentiating w.r.t. 'x' we get,
\(\frac { dy }{ dx } \) = m(1)+0 ⇒ m = \(\frac { dy }{ dx } \) ...(2)
Substituting (2) in (1) we get,
y = \(\left( \frac { dy }{ dx } \right) x+\frac { a }{ \frac { dy }{ dx } } \Rightarrow y=\frac { \left( \frac { dy }{ dx } \right) ^{ 2 }x+a }{ \left( \frac { dy }{ dx } \right) } \)
⇒ y\(\left( \frac { dy }{ dx } \right) =x\left( \frac { dy }{ dx } \right) ^{ 2 }\)+ a which is the required differential equation.