60.2.370 problem 947

Internal problem ID [10957]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 947
Date solved : Monday, January 27, 2025 at 10:31:37 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=\frac {2 x^{2} \cos \left (x \right )+2 \sin \left (x \right ) x^{3}-2 x \sin \left (x \right )+2 x +2 x^{2} y^{2}-4 y \sin \left (x \right ) x +4 y \cos \left (x \right ) x^{2}+4 y x +3-\cos \left (2 x \right )-2 \sin \left (2 x \right ) x -4 \sin \left (x \right )+x^{2} \cos \left (2 x \right )+x^{2}+4 \cos \left (x \right ) x}{2 x^{3}} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 44

dsolve(diff(y(x),x) = 1/2*(2*x^2*cos(x)+2*sin(x)*x^3-2*x*sin(x)+2*x+2*x^2*y(x)^2-4*y(x)*sin(x)*x+4*y(x)*cos(x)*x^2+4*x*y(x)+3-cos(2*x)-2*sin(2*x)*x-4*sin(x)+x^2*cos(2*x)+x^2+4*cos(x)*x)/x^3,y(x), singsol=all)
 

\[ y = \frac {\left (\cos \left (x \right ) x -\sin \left (x \right )+1\right ) \ln \left (x \right )-\cos \left (x \right ) c_{1} x +\sin \left (x \right ) c_{1} +x -c_{1}}{x \left (-\ln \left (x \right )+c_{1} \right )} \]

✓ Solution by Mathematica

Time used: 0.561 (sec). Leaf size: 45

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

\begin{align*} y(x)\to \frac {\sin (x)-x \cos (x)-1}{x}+\frac {1}{-\log (x)+c_1} \\ y(x)\to \frac {\sin (x)-x \cos (x)-1}{x} \\ \end{align*}