89.2.12 problem 12

Internal problem ID [24277]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 12
Date solved : Thursday, October 02, 2025 at 10:05:49 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.103 (sec). Leaf size: 54

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

\[ y = \left ({\mathrm e}^{\operatorname {RootOf}\left (\ln \left (x^{4} \left ({\mathrm e}^{\textit {\_Z}}-1\right )^{3}\right ) {\mathrm e}^{\textit {\_Z}}+4 c_1 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-\ln \left (x^{4} \left ({\mathrm e}^{\textit {\_Z}}-1\right )^{3}\right )-4 c_1 -\textit {\_Z} -1\right )}-1\right ) x \]

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 37

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

\[ \text {Solve}\left [-\frac {x}{y(x)}+3 \log \left (\frac {y(x)}{x}\right )+\log \left (\frac {y(x)}{x}+1\right )=-4 \log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.818 (sec). Leaf size: 24

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

\[ \log {\left (y{\left (x \right )} \right )} = C_{1} + \frac {x}{4 y{\left (x \right )}} - \log {\left (\sqrt [4]{\frac {x}{y{\left (x \right )}} + 1} \right )} \]