9.19.18 problem 18

Internal problem ID [3302]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 06:33:01 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.212 (sec). Leaf size: 21

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

\begin{align*} y &= \frac {x^{4}}{8} \\ y &= c_{1} \left (x^{2}-2 c_{1} \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.948 (sec). Leaf size: 142

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

\begin{align*} \text {Solve}\left [\frac {1}{4} \log (y(x))-\frac {\sqrt {x^6-8 x^2 y(x)} \text {arctanh}\left (\frac {\sqrt {x^4-8 y(x)}}{x^2}\right )}{2 x \sqrt {x^4-8 y(x)}}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {\sqrt {x^6-8 x^2 y(x)} \text {arctanh}\left (\frac {\sqrt {x^4-8 y(x)}}{x^2}\right )}{2 x \sqrt {x^4-8 y(x)}}+\frac {1}{4} \log (y(x))=c_1,y(x)\right ]\\ y(x)&\to \frac {x^4}{8} \end{align*}

✓ Sympy. Time used: 1.234 (sec). Leaf size: 12

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

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