67.17.42 problem 42

Internal problem ID [16863]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 25. Review exercises for part III. page 447
Problem number : 42
Date solved : Thursday, October 02, 2025 at 01:39:51 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 43

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

\begin{align*} y &= \int \frac {x}{\sqrt {2 x^{3}-c_1}}d x +c_2 \\ y &= -\int \frac {x}{\sqrt {2 x^{3}-c_1}}d x +c_2 \\ \end{align*}

✓ Mathematica. Time used: 1.223 (sec). Leaf size: 195

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

\begin{align*} y(x)&\to c_2-\frac {x^2 \sqrt {1+\frac {2 x^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {2 x^3}{c_1}\right )}{2 \sqrt {2 x^3+c_1}}\\ y(x)&\to \frac {x^2 \sqrt {1+\frac {2 x^3}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {2 x^3}{c_1}\right )}{2 \sqrt {2 x^3+c_1}}+c_2\\ y(x)&\to c_2\\ y(x)&\to -\frac {3 \sqrt {x^3} \operatorname {Gamma}\left (\frac {5}{3}\right )}{\sqrt {2} x \operatorname {Gamma}\left (\frac {2}{3}\right )}+c_2\\ y(x)&\to \frac {3 \sqrt {x^3} \operatorname {Gamma}\left (\frac {5}{3}\right )}{\sqrt {2} x \operatorname {Gamma}\left (\frac {2}{3}\right )}+c_2 \end{align*}

✓ Sympy. Time used: 163.310 (sec). Leaf size: 112

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

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