60.1.391 problem 402

Internal problem ID [10405]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 402
Date solved : Sunday, March 30, 2025 at 04:37:09 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 93

ode:=3*diff(y(x),x)^2+4*x*diff(y(x),x)-y(x)+x^2 = 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: 3.327 (sec). Leaf size: 121

ode=x^2 - y[x] + 4*x*D[y[x],x] + 3*D[y[x],x]^2==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: 1.624 (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)
 

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