Internal problem ID [3645]
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]
\[ \boxed {\left (-\sqrt {x^{2}+1}+x \right ) y^{\prime }-y-\sqrt {1+y^{2}}=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ c_{1} +x^{2}+x \sqrt {x^{2}+1}+\operatorname {arcsinh}\left (x \right )+y \left (x \right ) \sqrt {y \left (x \right )^{2}+1}+\operatorname {arcsinh}\left (y \left (x \right )\right )-y \left (x \right )^{2} = 0 \]
✓ Solution by Mathematica
Time used: 0.922 (sec). Leaf size: 84
DSolve[(x-Sqrt[1+x^2])y'[x]==y[x]+Sqrt[1+ y[x]^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )\right )\&\right ]\left [\frac {1}{2} \left (\log \left (\sqrt {x^2+1}-x\right )-x \left (\sqrt {x^2+1}+x\right )\right )+c_1\right ] \]