3.5.3.25 Example 25

\begin{equation} y=x\left ( p+a\sqrt {1+p^{2}}\right ) \tag {1}\end{equation}

Comparing the above to

\begin{equation} y=xf\left ( p\right ) +g\left ( p\right ) \tag {2}\end{equation}

Shows that

\begin{align*} f & =p+a\sqrt {1+p^{2}}\\ g & =0 \end{align*}

Since \(f\left ( p\right ) \neq p\) then this is d’Almbert ode. Taking derivative of (2) w.r.t. \(x\) gives

\begin{align} p & =\frac {d}{dx}\left ( xf\left ( p\right ) \right ) \nonumber \\ p & =f\left ( p\right ) +xf^{\prime }\left ( p\right ) \frac {dp}{dx}\nonumber \\ p & =\left ( p+a\sqrt {1+p^{2}}\right ) +x\left ( 1+\frac {ap}{\sqrt {1+p^{2}}}\right ) \frac {dp}{dx}\nonumber \\ -a\sqrt {1+p^{2}} & =x\left ( 1+\frac {ap}{\sqrt {1+p^{2}}}\right ) \frac {dp}{dx}\tag {3}\end{align}

Singular solution is when \(\frac {dp}{dx}=0\) which results in

\[ -a\sqrt {1+p^{2}}=0 \]

Hence \(p=\pm i\). None of these satisfy the ode. Hence not used. To find general solution, we need to solve (3) for \(p\). EQ (3) becomes

\begin{align*} \frac {dp}{dx} & =\frac {-a\sqrt {1+p^{2}}}{x\left ( 1+\frac {ap}{\sqrt {1+p^{2}}}\right ) }\\ & =\frac {-a\sqrt {1+p^{2}}\sqrt {1+p^{2}}}{x\left ( \sqrt {1+p^{2}}+ap\right ) }\\ & =\frac {-a\left ( 1+p^{2}\right ) }{x\left ( \sqrt {1+p^{2}}+ap\right ) }\end{align*}

This is separable ode.

\begin{align} \frac {dp}{dx}\left ( \frac {\sqrt {1+p^{2}}+ap}{1+p^{2}}\right ) & =-\frac {a}{x}\nonumber \\ \int \left ( \frac {\sqrt {1+p^{2}}+ap}{1+p^{2}}\right ) dp & =-\int \frac {a}{x}dx\nonumber \\ \operatorname {arcsinh}\left ( p\right ) +\frac {a}{2}\ln \left ( 1+p^{2}\right ) & =-a\ln x+c\nonumber \\ \ln x+\frac {1}{a}\operatorname {arcsinh}\left ( p\right ) +\frac {1}{2}\ln \left ( 1+p^{2}\right ) +c & =0\nonumber \\ xe^{\frac {1}{a}\operatorname {arcsinh}\left ( p\right ) }\sqrt {1+p^{2}}c & =0\tag {4}\end{align}

From (1), we can solve for \(p\) which gives

\begin{align*} p_{1} & =-\frac {1}{x}\left ( \frac {\left ( ay+\sqrt {x^{2}+y^{2}-a^{2}x^{2}}\right ) a}{a^{2}-1}-y\right ) \\ p_{2} & =-\frac {1}{x}\left ( \frac {-\left ( -ay+\sqrt {x^{2}+y^{2}-a^{2}x^{2}}\right ) a}{a^{2}-1}-y\right ) \end{align*}

Substituting each of the above into (4), gives two implicit solutions

\begin{align*} xe^{\frac {1}{a}\operatorname {arcsinh}\left ( p_{1}\right ) }\sqrt {1+p_{1}^{2}}c & =0\\ xe^{\frac {1}{a}\operatorname {arcsinh}\left ( p_{2}\right ) }\sqrt {1+p_{2}^{2}}c & =0 \end{align*}