77.1.96 problem 123 (page 179)

Internal problem ID [17915]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 123 (page 179)
Date solved : Monday, March 31, 2025 at 04:51:20 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✗ Maple

ode:=a^2*diff(diff(y(x),x),x) = 2*x*(1+diff(y(x),x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.481 (sec). Leaf size: 50

ode=a^2*D[y[x],{x,2}]==2*x*Sqrt[1+D[y[x],x]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2-\frac {1}{4} \sqrt {\pi } a \left (\text {erf}\left (\frac {x}{a}\right ) (\cosh (c_1)-\sinh (c_1))-\text {erfi}\left (\frac {x}{a}\right ) (\cosh (c_1)+\sinh (c_1))\right ) \]

✗ Sympy

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

Timed Out