56.1.62 problem 62

Internal problem ID [8774]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 62
Date solved : Monday, January 27, 2025 at 04:50:07 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 51

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

\begin{align*} \int _{}^{y}\frac {1}{\sqrt {2 \ln \left (\textit {\_a} \right )-c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {1}{\sqrt {2 \ln \left (\textit {\_a} \right )-c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.070 (sec). Leaf size: 93

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

\begin{align*} y(x)\to \exp \left (-\text {erf}^{-1}\left (-i \sqrt {\frac {2}{\pi }} \sqrt {e^{c_1} (x+c_2){}^2}\right ){}^2-\frac {c_1}{2}\right ) \\ y(x)\to \exp \left (-\text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} \sqrt {e^{c_1} (x+c_2){}^2}\right ){}^2-\frac {c_1}{2}\right ) \\ \end{align*}