60.2.392 problem 969

Internal problem ID [10979]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 969
Date solved : Monday, January 27, 2025 at 10:37:41 PM
CAS classification : [[_homogeneous, `class D`]]

\begin{align*} y^{\prime }&=\frac {y \sin \left (\frac {3 y}{2 x}\right ) \cos \left (\frac {y}{2 x}\right ) x +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 ) x +y \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right )-\sin \left (\frac {y}{x}\right ) y x -y \sin \left (\frac {y}{x}\right )+2 \sin \left (\frac {y}{x}\right ) \cos \left (\frac {y}{2 x}\right ) \sin \left (\frac {y}{2 x}\right ) x}{2 \cos \left (\frac {y}{x}\right ) \sin \left (\frac {y}{2 x}\right ) x \cos \left (\frac {y}{2 x}\right ) \left (1+x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.042 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 60.255 (sec). Leaf size: 29

DSolve[D[y[x],x] == (Csc[y[x]/(2*x)]*Sec[y[x]/(2*x)]*Sec[y[x]/x]*(x*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 + (x*Cos[y[x]/(2*x)]*Sin[y[x]/(2*x)]*y[x])/2 - (Sin[y[x]/x]*y[x])/2 - (x*Sin[y[x]/x]*y[x])/2 + (Cos[y[x]/(2*x)]*Sin[(3*y[x])/(2*x)]*y[x])/2 + (x*Cos[y[x]/(2*x)]*Sin[(3*y[x])/(2*x)]*y[x])/2))/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x \arcsin \left (\exp \left (\int _1^x\frac {1}{K[1]^2+K[1]}dK[1]+c_1\right )\right ) \]