The differential equation representing the family of curves $y^2=2 c(x+\sqrt{c})$, where $c$ $>0$, is a…
The differential equation representing the family of curves $y^2=2 c(x+\sqrt{c})$, where $c$ $>0$, is a parameter, is of order and degree as follows:
order 1 , degree 2
order 1, degree 1
order 1, degree 3
order 2, degree 2
Solution
$
\begin{aligned}
& y^2=2 c(x+\sqrt{c} ) .....(i)\\
& 2 y y^{\prime}=2 c \cdot 1 \text { or } y y^{\prime}=c ....(ii)\\
& \Rightarrow y^2=2 y y^{\prime}\left(x+\sqrt{y y^{\prime}}\right)
\end{aligned}
$
[on putting value of c from (ii) in (i)]
On simplifying, we get
$
\left(y-2 x y^{\prime}\right)^2=4 y y^{\prime 3} .....(iii)
$
Hence equation (iii) is of order 1 and degree 3