29.25.31 problem 728

Internal problem ID [5318]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 728
Date solved : Monday, January 27, 2025 at 11:10:40 AM
CAS classification : [_separable]

\begin{align*} \left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{{3}/{2}} y^{\prime }&=1+y^{2} \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 28

dsolve((y(x)+sqrt(1+y(x)^2))*(x^2+1)^(3/2)*diff(y(x),x) = 1+y(x)^2,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.513 (sec). Leaf size: 115

DSolve[(1+x^2)^(3/2)*(y[x]+Sqrt[1+y[x]^2])*D[y[x],x]==1+y[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {i \left (1+e^{\frac {x}{\sqrt {x^2+1}}+c_1}\right )}{\sqrt {1+2 e^{\frac {x}{\sqrt {x^2+1}}+c_1}}} \\ y(x)\to \frac {i \left (1+e^{\frac {x}{\sqrt {x^2+1}}+c_1}\right )}{\sqrt {1+2 e^{\frac {x}{\sqrt {x^2+1}}+c_1}}} \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}