60.2.383 problem 960

Internal problem ID [10970]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 960
Date solved : Monday, January 27, 2025 at 10:35:37 PM
CAS classification : [[_homogeneous, `class D`]]

\begin{align*} y^{\prime }&=\frac {-y \sin \left (\frac {y}{x}\right )+y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )+y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )+2 \sin \left (\frac {y}{x}\right ) x^{2} \sin \left (\frac {y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right )}{2 \cos \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 16

dsolve(diff(y(x),x) = 1/2*(-y(x)*sin(y(x)/x)+y(x)*sin(3/2*y(x)/x)*cos(1/2*y(x)/x)+y(x)*cos(1/2*y(x)/x)*sin(1/2*y(x)/x)+2*sin(y(x)/x)*x^2*sin(1/2*y(x)/x)*cos(1/2*y(x)/x))/cos(y(x)/x)/cos(1/2*y(x)/x)/sin(1/2*y(x)/x)/x,y(x), singsol=all)
 

\[ y = \frac {\arccos \left ({\mathrm e}^{2 x} c_{1} +1\right ) x}{2} \]

✓ Solution by Mathematica

Time used: 41.348 (sec). Leaf size: 19

DSolve[D[y[x],x] == (Csc[y[x]/(2*x)]*Sec[y[x]/(2*x)]*Sec[y[x]/x]*(x^2*Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)]*Sin[y[x]/x] + (Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)]*y[x])/2 - (Sin[y[x]/x]*y[x])/2 + (Cos[y[x]/(2*x)]*Sin[(3*y[x])/(2*x)]*y[x])/2))/x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to x \arcsin \left (e^{x+c_1}\right ) \\ y(x)\to 0 \\ \end{align*}