29.18.23 problem 499

Internal problem ID [5097]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 18
Problem number : 499
Date solved : Monday, January 27, 2025 at 10:10:56 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

\begin{align*} \left (a x +b y\right ) y^{\prime }+x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.183 (sec). Leaf size: 63

dsolve((a*x+b*y(x))*diff(y(x),x)+x = 0,y(x), singsol=all)
 

\[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z}^{2} b -{\mathrm e}^{\operatorname {RootOf}\left (\left (4 \,{\mathrm e}^{\textit {\_Z}} b {\cosh \left (\frac {\sqrt {a^{2}-4 b}\, \left (2 c_{1} +\textit {\_Z} +2 \ln \left (x \right )\right )}{2 a}\right )}^{2}+a^{2}-4 b \right ) x^{2}\right )}+1+\textit {\_Z} a \right ) x \]

✓ Solution by Mathematica

Time used: 0.123 (sec). Leaf size: 75

DSolve[(a x+b y[x])D[y[x],x]+x==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {a \arctan \left (\frac {a+\frac {2 b y(x)}{x}}{\sqrt {4 b-a^2}}\right )}{\sqrt {4 b-a^2}}+\frac {1}{2} \log \left (\frac {a y(x)}{x}+\frac {b y(x)^2}{x^2}+1\right )=-\log (x)+c_1,y(x)\right ] \]