29.29.28 problem 850

Internal problem ID [5433]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 850
Date solved : Sunday, March 30, 2025 at 08:13:10 AM
CAS classification : [_rational, _dAlembert]

\begin{align*} x {y^{\prime }}^{2}+4 y^{\prime }-2 y&=0 \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 67

ode:=x*diff(y(x),x)^2+4*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2 \,{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} x +4 \,{\mathrm e}^{\textit {\_Z}} x -4 \,{\mathrm e}^{\textit {\_Z}}+c_1 +8 \textit {\_Z} -4 x \right )} x +4 \operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}} x +4 \,{\mathrm e}^{\textit {\_Z}} x -4 \,{\mathrm e}^{\textit {\_Z}}+c_1 +8 \textit {\_Z} -4 x \right )+\frac {c_1}{2}-2 x \]

✓ Mathematica. Time used: 30.514 (sec). Leaf size: 90

ode=x (D[y[x],x])^2+4 D[y[x],x]-2 y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\left \{x=-\frac {2 (2 K[1]-y(K[1]))}{K[1]^2},y(x)=4 \left (\frac {2}{K[1]}+\log (K[1])\right ) \exp \left (-4 \left (\frac {1}{2} \log (2-K[1])-\frac {1}{2} \log (K[1])\right )\right )+c_1 \exp \left (-4 \left (\frac {1}{2} \log (2-K[1])-\frac {1}{2} \log (K[1])\right )\right )\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x)**2 - 2*y(x) + 4*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(2*x*y(x) + 4) - 2)/x cannot be solved by the factorable group method