29.26.11 problem 747

Internal problem ID [5332]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 26
Problem number : 747
Date solved : Monday, January 27, 2025 at 11:15:32 AM
CAS classification : [_separable]

\begin{align*} y^{\prime } \left (1+\sinh \left (x \right )\right ) \sinh \left (y\right )+\cosh \left (x \right ) \left (\cosh \left (y\right )-1\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 123

dsolve(diff(y(x),x)*(1+sinh(x))*sinh(y(x))+cosh(x)*(cosh(y(x))-1) = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= -2 \,\operatorname {arctanh}\left (\frac {c_{1} \sqrt {2}\, \sqrt {\frac {-{\mathrm e}^{2 x} {\mathrm e}^{x} c_{1} +\left (-2 c_{1} +2\right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} c_{1}}{c_{1}^{2}}}}{c_{1} {\mathrm e}^{2 x}+\left (2 c_{1} -2\right ) {\mathrm e}^{x}-c_{1}}\right ) \\ y \left (x \right ) &= 2 \,\operatorname {arctanh}\left (\frac {c_{1} \sqrt {2}\, \sqrt {\frac {-{\mathrm e}^{2 x} {\mathrm e}^{x} c_{1} +\left (-2 c_{1} +2\right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} c_{1}}{c_{1}^{2}}}}{c_{1} {\mathrm e}^{2 x}+\left (2 c_{1} -2\right ) {\mathrm e}^{x}-c_{1}}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.333 (sec). Leaf size: 32

DSolve[D[y[x],x]*(1+Sinh[x])*Sinh[y[x]]+Cosh[x]*(Cosh[y[x]]-1)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 0 \\ y(x)\to 2 \text {arcsinh}\left (\frac {c_1}{4 \sqrt {\sinh (x)+1}}\right ) \\ y(x)\to 0 \\ \end{align*}