29.37.29 problem 1152

Internal problem ID [5687]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 37
Problem number : 1152
Date solved : Sunday, March 30, 2025 at 10:01:52 AM
CAS classification : [_Clairaut]

\begin{align*} y^{\prime } \ln \left (y^{\prime }+\sqrt {a +{y^{\prime }}^{2}}\right )-\sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y&=0 \end{align*}

✗ Maple

ode:=diff(y(x),x)*ln(diff(y(x),x)+(a+diff(y(x),x)^2)^(1/2))-(1+diff(y(x),x)^2)^(1/2)-x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 60.045 (sec). Leaf size: 38

ode=D[y[x],x]*Log[D[y[x],x]+Sqrt[a+(D[y[x],x])^2]]-Sqrt[1+(D[y[x],x])^2]-x*D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -c_1 \log \left (\sqrt {a+c_1{}^2}+c_1\right )+c_1 x+\sqrt {1+c_1{}^2} \]

✗ Sympy

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

NotImplementedError : multiple generators [_X0, log(_X0 + sqrt(_X0**2 + a)), sqrt(_X0**2 + 1)] 
No algorithms are implemented to solve equation -_X0*x + _X0*log(_X0 + sqrt(_X0**2 + a)) - sqrt(_X0**2 + 1) + y(x)