56.1.62 problem 62

Internal problem ID [8774]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 62
Date solved : Sunday, March 30, 2025 at 01:32:55 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 51

ode:=y(x)*diff(diff(y(x),x),x) = 1; 
dsolve(ode,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*}

✓ Mathematica. Time used: 60.07 (sec). Leaf size: 93

ode=y[x]*D[y[x],{x,2}]==1; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 39.041 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), (x, 2)) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {C_{1} + 2 \log {\left (u \right )}}}\, du = C_{2} + x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {C_{1} + 2 \log {\left (u \right )}}}\, du = C_{2} - x\right ] \]