73.17.42 problem 42

Internal problem ID [15576]
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 : Tuesday, January 28, 2025 at 08:02:34 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.098 (sec). Leaf size: 43

dsolve(x*diff(y(x),x$2)-diff(y(x),x)=-3*x* diff(y(x),x)^3,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*}

✓ Solution by Mathematica

Time used: 1.687 (sec). Leaf size: 195

DSolve[x*D[y[x],{x,2}]-D[y[x],x]==-3*x* (D[y[x],x])^3,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*}