23.2.87 problem 89

Internal problem ID [5442]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 89
Date solved : Tuesday, September 30, 2025 at 12:43:25 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} 3 {y^{\prime }}^{2}+4 x y^{\prime }+x^{2}-y&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 93

ode:=3*diff(y(x),x)^2+4*x*diff(y(x),x)+x^2-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {x^{2}}{3} \\ y &= -\frac {x^{2}}{4}+\frac {\sqrt {3}\, c_1 x}{6}+\frac {c_1^{2}}{4} \\ y &= -\frac {x^{2}}{4}-\frac {\sqrt {3}\, c_1 x}{6}+\frac {c_1^{2}}{4} \\ y &= -\frac {x^{2}}{4}-\frac {\sqrt {3}\, c_1 x}{6}+\frac {c_1^{2}}{4} \\ y &= -\frac {x^{2}}{4}+\frac {\sqrt {3}\, c_1 x}{6}+\frac {c_1^{2}}{4} \\ \end{align*}

✓ Mathematica. Time used: 2.992 (sec). Leaf size: 121

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

\begin{align*} y(x)&\to \frac {1}{12} \left (-3 x^2+2 x-2 e^{c_1} (x+1)+1+e^{2 c_1}\right )\\ y(x)&\to \frac {-3 x^2-3 x^2 \tanh ^2\left (\frac {c_1}{2}\right )+4 x+2 (3 x-2) x \tanh \left (\frac {c_1}{2}\right )+4}{12 \left (-1+\tanh \left (\frac {c_1}{2}\right )\right ){}^2}\\ y(x)&\to -\frac {x^2}{3}\\ y(x)&\to \frac {1}{12} \left (-3 x^2+2 x+1\right ) \end{align*}

✓ Sympy. Time used: 1.015 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = - \frac {x^{2}}{3} + \frac {\left (C_{1} + x\right )^{2}}{12} \]