84.2.6 problem 1.12
Internal
problem
ID
[22075]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
1.
Basic
concepts.
Supplementary
problems
Problem
number
:
1.12
Date
solved
:
Friday, October 03, 2025 at 07:59:45 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
\begin{align*} {y^{\prime \prime }}^{{3}/{2}}+y&=x \end{align*}
✓ Maple. Time used: 0.092 (sec). Leaf size: 245
ode:=diff(diff(y(x),x),x)^(3/2)+y(x) = x;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= x \\
y &= x +\operatorname {RootOf}\left (-x -5 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {30 \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}}-25 c_1 +25}}d \textit {\_f} +c_2 \right ) \\
y &= x +\operatorname {RootOf}\left (-x +5 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {30 \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}}-25 c_1 +25}}d \textit {\_f} +c_2 \right ) \\
y &= x +\operatorname {RootOf}\left (-x -5 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {25+15 i \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}} \sqrt {3}-15 \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}}-25 c_1}}d \textit {\_f} +c_2 \right ) \\
y &= x +\operatorname {RootOf}\left (-x +5 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {25+15 i \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}} \sqrt {3}-15 \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}}-25 c_1}}d \textit {\_f} +c_2 \right ) \\
y &= x +\operatorname {RootOf}\left (-x -5 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {25-15 i \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}} \sqrt {3}-15 \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}}-25 c_1}}d \textit {\_f} +c_2 \right ) \\
y &= x +\operatorname {RootOf}\left (-x +5 \int _{}^{\textit {\_Z}}\frac {1}{\sqrt {25-15 i \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}} \sqrt {3}-15 \textit {\_f} \left (-\textit {\_f} \right )^{{2}/{3}}-25 c_1}}d \textit {\_f} +c_2 \right ) \\
\end{align*}
✓ Mathematica. Time used: 2.777 (sec). Leaf size: 202
ode=D[y[x],{x,2}]^(3/2)+y[x]==x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solve}\left [-\sqrt {15} x-\frac {15 (x-y(x)) \sqrt {1-\frac {18 (x-y(x))^{5/3}}{15+20 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{5},\frac {8}{5},\frac {18 (x-y(x))^{5/3}}{20 c_1+15}\right )}{\sqrt {-18 (x-y(x))^{5/3}+15+20 c_1}}=c_2,y(x)\right ]\\ \text {Solve}\left [\sqrt {15} x-\frac {15 (x-y(x)) \sqrt {1-\frac {18 (x-y(x))^{5/3}}{15+20 c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{5},\frac {8}{5},\frac {18 (x-y(x))^{5/3}}{20 c_1+15}\right )}{\sqrt {-18 (x-y(x))^{5/3}+15+20 c_1}}=c_2,y(x)\right ] \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + y(x) + Derivative(y(x), (x, 2))**(3/2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve -x + y(x) + Derivative(y(x), (x, 2))**(3/2)