75.9.5 problem 224

Internal problem ID [16771]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number : 224
Date solved : Monday, March 31, 2025 at 03:16:48 PM
CAS classification : [_dAlembert]

\begin{align*} y&=\frac {3 x y^{\prime }}{2}+{\mathrm e}^{y^{\prime }} \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 153

ode:=y(x) = 3/2*x*diff(y(x),x)+exp(diff(y(x),x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 1 \\ \frac {x^{3} c_1}{{\left (3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-2 y\right )}^{3}}+x -\frac {54 \left (\operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )^{2} x^{2}+2 x \left (x -\frac {2 y}{3}\right ) \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )+2 x^{2}-\frac {4 x y}{3}+\frac {4 y^{2}}{9}\right ) x \,{\mathrm e}^{-\frac {3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-2 y}{3 x}}}{{\left (3 x \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {2 y}{3 x}}}{3 x}\right )-2 y\right )}^{3}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.519 (sec). Leaf size: 52

ode=y[x]==3/2*x*D[y[x],x]+Exp[D[y[x],x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\left \{x=-\frac {2 e^{K[1]} \left (K[1]^2-2 K[1]+2\right )}{K[1]^3}+\frac {c_1}{K[1]^3},y(x)=\frac {3}{2} x K[1]+e^{K[1]}\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*Derivative(y(x), x)/2 + y(x) - exp(Derivative(y(x), x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE LambertW(-2*(-1)**(1/3)*exp(2*y(x)/(3*x))/(3*x)) + Derivative(y(x), x) - 2*y(x)/(3*x) cannot be solved by the factorable group method