75.14.31 problem 357

Internal problem ID [16945]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 14. Differential equations admitting of depression of their order. Exercises page 107
Problem number : 357
Date solved : Tuesday, January 28, 2025 at 09:42:58 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y y^{\prime \prime }&={y^{\prime }}^{2}+1 \end{align*}

✓ Solution by Maple

Time used: 0.158 (sec). Leaf size: 55

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

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

✓ Solution by Mathematica

Time used: 0.955 (sec). Leaf size: 464

DSolve[y[x]*D[y[x],{x,2}]==1+D[y[x],x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 (-c_1)}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 (-c_1)}}\right )}{\sqrt {-e^{2 (-c_1)}} \sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {1-\text {$\#$1}^2 e^{2 c_1}} \text {arcsinh}\left (\text {$\#$1} \sqrt {-e^{2 c_1}}\right )}{\sqrt {-e^{2 c_1}} \sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}}\&\right ][x+c_2] \\ \end{align*}