60.1.29 problem Problem 42

Internal problem ID [15157]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 42
Date solved : Thursday, October 02, 2025 at 10:06:09 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }&=\left (x -5 y\right )^{{1}/{3}}+2 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 80

ode:=diff(y(x),x) = (x-5*y(x))^(1/3)+2; 
dsolve(ode,y(x), singsol=all);
 

\[ x +\frac {81 \ln \left (729-625 y+125 x \right )}{125}-\frac {27 \left (x -5 y\right )^{{1}/{3}}}{25}+\frac {162 \ln \left (9+5 \left (x -5 y\right )^{{1}/{3}}\right )}{125}-\frac {81 \ln \left (25 \left (x -5 y\right )^{{2}/{3}}-45 \left (x -5 y\right )^{{1}/{3}}+81\right )}{125}+\frac {3 \left (x -5 y\right )^{{2}/{3}}}{10}-c_1 = 0 \]

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 70

ode=D[y[x],x]==(x-5*y[x])^(1/3)+2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [5 y(x)+5 \left (-y(x)+\frac {3}{50} (x-5 y(x))^{2/3}-\frac {27}{125} \sqrt [3]{x-5 y(x)}+\frac {243}{625} \log \left (5 \sqrt [3]{x-5 y(x)}+9\right )+\frac {x}{5}\right )=c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.055 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(x - 5*y(x))**(1/3) + Derivative(y(x), x) - 2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + x + \frac {3 \cdot 5^{\frac {2}{3}} \left (\frac {x}{5} - y{\left (x \right )}\right )^{\frac {2}{3}}}{10} - \frac {27 \sqrt [3]{5} \sqrt [3]{\frac {x}{5} - y{\left (x \right )}}}{25} + \frac {243 \log {\left (5 \sqrt [3]{5} \sqrt [3]{\frac {x}{5} - y{\left (x \right )}} + 9 \right )}}{125} = 0 \]