# Using substitution p=y', reduce the differential equation y''=1+(y')^2 to first order and solve.

## Question:

Using substitution p=y', reduce the differential equation {eq}y''=1+(y')^2 to {/eq} first order and solve.

## Method of Variables Separable:

Converting or separating equation of the form {eq}m(x,y)dx + n(x,y)dy = 0 {/eq} to {eq}u(x)dx + v(y)dy = 0 {/eq} is the method of of variable separable.

## Answer and Explanation:

Given that: {eq}\displaystyle y'' = 1 + {(y')^2} {/eq}

{eq}\displaystyle\ \eqalign{ & p' = 1 + {p^2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{Using substitution }}p = y'} \right) \cr & \frac{{dp}}{{dx}} = 1 + {p^2} \cr & \frac{{dp}}{{{p^2} + 1}} = dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{From method of variables separable}}} \right) \cr & \int {\frac{{dp}}{{{p^2} + 1}} = \int {dx\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{Taking integration both side}}} \right)} } \cr & {\tan ^{ - 1}}p = x + c\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\int {\frac{{dx}}{{{x^2} + {a^2}}} = \frac{1}{a}{{\tan }^{ - 1}}\left( {\frac{x}{a}} \right) + c,\int {dx = x + c} } } \right) \cr & p = \tan (x + c) \cr & p(x) = \tan (x + c) \cr} {/eq}