83.32.9 problem 9

Internal problem ID [19337]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (F) at page 113
Problem number : 9
Date solved : Monday, March 31, 2025 at 07:08:52 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.070 (sec). Leaf size: 34

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

\[ y = \int \operatorname {RootOf}\left (-\textit {\_Z} \sqrt {\textit {\_Z}^{2}+1}-\textit {\_Z}^{2}-\operatorname {arcsinh}\left (\textit {\_Z} \right )-2 \ln \left (x \right )+c_1 \right )d x +c_2 \]

✓ Mathematica. Time used: 0.176 (sec). Leaf size: 64

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

\[ y(x)\to \int _1^x\text {InverseFunction}\left [\frac {1}{2} \left (\text {$\#$1} \left (\text {$\#$1}+\sqrt {\text {$\#$1}^2+1}\right )-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )\right )\&\right ][c_1-\log (K[1])]dK[1]+c_2 \]

✗ Sympy

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

NotImplementedError : The given ODE -x*Derivative(y(x), (x, 2))/2 + Derivative(y(x), x) + 1/(2*x*Derivative(y(x), (x, 2))) cannot be solved by the factorable group method