83.4.5 problem 5

Internal problem ID [19077]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (C) at page 12
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 12:52:07 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} \left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y-\left (y \sin \left (\frac {y}{x}\right )-x \cos \left (\frac {y}{x}\right )\right ) x y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 18

dsolve((x*cos(y(x)/x)+y(x)*sin(y(x)/x) )*y(x) - (y(x)*sin(y(x)/x)-x*cos(y(x)/x))*x*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = x \operatorname {RootOf}\left (\textit {\_Z} \cos \left (\textit {\_Z} \right ) x^{2}-c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.320 (sec). Leaf size: 31

DSolve[(x*Cos[y[x]/x]+y[x]*Sin[y[x]/x] )*y[x] - (y[x]*Sin[y[x]/x]-x*Cos[y[x]/x])*x*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [-\log \left (\frac {y(x)}{x}\right )-\log \left (\cos \left (\frac {y(x)}{x}\right )\right )=2 \log (x)+c_1,y(x)\right ] \]