28.1.112 problem 135
Internal
problem
ID
[4418]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
2.
First-Order
and
Simple
Higher-Order
Differential
Equations.
Page
78
Problem
number
:
135
Date
solved
:
Monday, January 27, 2025 at 09:16:23 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
\begin{align*} \cos \left (y\right )+\sin \left (y\right ) \left (x -\sin \left (y\right ) \cos \left (y\right )\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.051 (sec). Leaf size: 240
dsolve(cos(y(x)) + sin(y(x)) * (x-sin(y(x))*cos(y(x)) )*diff(y(x),x)=0,y(x), singsol=all)
\[
y \left (x \right ) = \arctan \left (\frac {{\mathrm e}^{2 \operatorname {RootOf}\left (-x \,{\mathrm e}^{4 \textit {\_Z}}+2 c_{1} {\mathrm e}^{3 \textit {\_Z}}+2 \,{\mathrm e}^{3 \textit {\_Z}} \textit {\_Z} -2 \,{\mathrm e}^{3 \textit {\_Z}}-2 x \,{\mathrm e}^{2 \textit {\_Z}}+2 c_{1} {\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}-x \right )}-1}{{\mathrm e}^{2 \operatorname {RootOf}\left (-x \,{\mathrm e}^{4 \textit {\_Z}}+2 c_{1} {\mathrm e}^{3 \textit {\_Z}}+2 \,{\mathrm e}^{3 \textit {\_Z}} \textit {\_Z} -2 \,{\mathrm e}^{3 \textit {\_Z}}-2 x \,{\mathrm e}^{2 \textit {\_Z}}+2 c_{1} {\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}-x \right )}+1}, \frac {2 \,{\mathrm e}^{\operatorname {RootOf}\left (-x \,{\mathrm e}^{4 \textit {\_Z}}+2 c_{1} {\mathrm e}^{3 \textit {\_Z}}+2 \,{\mathrm e}^{3 \textit {\_Z}} \textit {\_Z} -2 \,{\mathrm e}^{3 \textit {\_Z}}-2 x \,{\mathrm e}^{2 \textit {\_Z}}+2 c_{1} {\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}-x \right )}}{{\mathrm e}^{2 \operatorname {RootOf}\left (-x \,{\mathrm e}^{4 \textit {\_Z}}+2 c_{1} {\mathrm e}^{3 \textit {\_Z}}+2 \,{\mathrm e}^{3 \textit {\_Z}} \textit {\_Z} -2 \,{\mathrm e}^{3 \textit {\_Z}}-2 x \,{\mathrm e}^{2 \textit {\_Z}}+2 c_{1} {\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}-x \right )}+1}\right )
\]
✓ Solution by Mathematica
Time used: 0.438 (sec). Leaf size: 26
DSolve[Cos[y[x]]+ Sin[y[x]] * (x-Sin[y[x]]*Cos[y[x]] )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}[x=\cos (y(x)) (\text {arctanh}(\sin (y(x)))-\sin (y(x)))+c_1 \cos (y(x)),y(x)]
\]