89.2.26 problem 28

Internal problem ID [24291]
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 : 28
Date solved : Thursday, October 02, 2025 at 10:07:44 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{2}+\left (x^{2}+3 y x +4 y^{2}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.306 (sec). Leaf size: 32

ode:=y(x)^2+(x^2+3*x*y(x)+4*y(x)^2)*diff(y(x),x) = 0; 
ic:=[y(2) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (1+8 \,{\mathrm e}^{\textit {\_Z}} \ln \left (x \right )-2 \,{\mathrm e}^{\textit {\_Z}}+4 \ln \left (x \right )+8 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+4 \textit {\_Z} \right )} x \]

✓ Mathematica. Time used: 0.142 (sec). Leaf size: 31

ode=( y[x]^2 )+(x^2+3*x*y[x]+4*y[x]^2 )*D[y[x],x]==0; 
ic={y[2]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.474 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 3*x*y(x) + 4*y(x)**2)*Derivative(y(x), x) + y(x)**2,0) 
ics = {y(2): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \log {\left (y{\left (x \right )} \right )} = - \frac {1}{4} + \frac {1}{\frac {x}{y{\left (x \right )}} + 2} \]