14.10 problem 391

Internal problem ID [3137]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 14
Problem number: 391.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_separable]

Solve \begin {gather*} \boxed {\left (-\sqrt {x^{2}+1}+x \right ) y^{\prime }-y-\sqrt {1+y^{2}}=0} \end {gather*}

Solution by Maple

Time used: 0.002 (sec). Leaf size: 39

dsolve((x-sqrt(x^2+1))*diff(y(x),x) = y(x)+sqrt(1+y(x)^2),y(x), singsol=all)
 

\[ c_{1}+x^{2}+x \sqrt {x^{2}+1}+\arcsinh \relax (x )+y \relax (x ) \sqrt {1+y \relax (x )^{2}}+\arcsinh \left (y \relax (x )\right )-y \relax (x )^{2} = 0 \]

Solution by Mathematica

Time used: 0.876 (sec). Leaf size: 79

DSolve[(x-Sqrt[1+x^2])y'[x]==y[x]+Sqrt[1+ y[x]^2],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )+\tanh ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\text {$\#$1}^2+1}}\right )\right )\&\right ]\left [-\frac {1}{2} x \left (\sqrt {x^2+1}+x\right )-\frac {1}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+1}}\right )+c_1\right ] \\ \end{align*}