82.19.5 problem Ex. 5
Internal
problem
ID
[18852]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
IV.
Singular
solutions.
problems
at
page
43
Problem
number
:
Ex.
5
Date
solved
:
Tuesday, January 28, 2025 at 12:30:21 PM
CAS
classification
:
[[_homogeneous, `class C`], _dAlembert]
\begin{align*} \left (1+y^{\prime }\right )^{3}&=\frac {7 \left (x +y\right ) \left (1-y^{\prime }\right )^{3}}{4 a} \end{align*}
✓ Solution by Maple
Time used: 1.011 (sec). Leaf size: 262
dsolve((1+diff(y(x),x))^3=2/8*7/a*(x+y(x))*(1-diff(y(x),x))^3,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -x \\
y \left (x \right ) &= -x +\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}-\frac {\left (\textit {\_a} \,a^{2}\right )^{{1}/{3}} \left (7 \textit {\_a} +4 a \right )}{a \textit {\_a} 14^{{2}/{3}}-2 \,14^{{1}/{3}} \left (a^{4} \textit {\_a}^{2}\right )^{{1}/{3}}-7 \left (\textit {\_a} \,a^{2}\right )^{{1}/{3}} \textit {\_a}}d \textit {\_a} +2 c_{1} \right ) \\
y \left (x \right ) &= -x +\operatorname {RootOf}\left (-x -\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a} \,a^{2}\right )^{{1}/{3}} \left (7 \textit {\_a} +4 a \right )}{i \sqrt {3}\, 14^{{2}/{3}} a \textit {\_a} +2 i \sqrt {3}\, 14^{{1}/{3}} \left (a^{4} \textit {\_a}^{2}\right )^{{1}/{3}}-a \textit {\_a} 14^{{2}/{3}}+2 \,14^{{1}/{3}} \left (a^{4} \textit {\_a}^{2}\right )^{{1}/{3}}-14 \left (\textit {\_a} \,a^{2}\right )^{{1}/{3}} \textit {\_a}}d \textit {\_a} +c_{1} \right ) \\
y \left (x \right ) &= -x +\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a} \,a^{2}\right )^{{1}/{3}} \left (7 \textit {\_a} +4 a \right )}{i \sqrt {3}\, 14^{{2}/{3}} a \textit {\_a} +a \textit {\_a} 14^{{2}/{3}}+2 i \sqrt {3}\, 14^{{1}/{3}} \left (a^{4} \textit {\_a}^{2}\right )^{{1}/{3}}-2 \,14^{{1}/{3}} \left (a^{4} \textit {\_a}^{2}\right )^{{1}/{3}}+14 \left (\textit {\_a} \,a^{2}\right )^{{1}/{3}} \textit {\_a}}d \textit {\_a} +c_{1} \right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 155.284 (sec). Leaf size: 54117
DSolve[(1+D[y[x],x])^3==2/8*7/a*(x+y[x])*(1-D[y[x],x])^3,y[x],x,IncludeSingularSolutions -> True]
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