60.2.386 problem 963

Internal problem ID [10973]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 963
Date solved : Monday, January 27, 2025 at 10:35:58 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Abel]

\begin{align*} y^{\prime }&=\frac {-4 \cos \left (x \right ) x +4 \sin \left (x \right ) x^{2}+4 x +4+4 y^{2}+8 y \cos \left (x \right ) x -8 y x +2 x^{2} \cos \left (2 x \right )+6 x^{2}-8 x^{2} \cos \left (x \right )+4 y^{3}+12 y^{2} \cos \left (x \right ) x -12 x y^{2}+6 y x^{2} \cos \left (2 x \right )+18 x^{2} y-24 y \cos \left (x \right ) x^{2}+x^{3} \cos \left (3 x \right )+15 x^{3} \cos \left (x \right )-6 x^{3} \cos \left (2 x \right )-10 x^{3}}{4 x} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 39

dsolve(diff(y(x),x) = 1/4*(-4*cos(x)*x+4*sin(x)*x^2+4*x+4+4*y(x)^2+8*y(x)*cos(x)*x-8*x*y(x)+2*x^2*cos(2*x)+6*x^2-8*x^2*cos(x)+4*y(x)^3+12*y(x)^2*cos(x)*x-12*x*y(x)^2+6*y(x)*x^2*cos(2*x)+18*x^2*y(x)-24*y(x)*cos(x)*x^2+x^3*cos(3*x)+15*x^3*cos(x)-6*x^3*cos(2*x)-10*x^3)/x,y(x), singsol=all)
 

\[ y = -\cos \left (x \right ) x +x -\frac {1}{3}+\frac {29 \operatorname {RootOf}\left (-81 \left (\int _{}^{\textit {\_Z}}\frac {1}{841 \textit {\_a}^{3}-27 \textit {\_a} +27}d \textit {\_a} \right )+\ln \left (x \right )+3 c_{1} \right )}{9} \]

✓ Solution by Mathematica

Time used: 0.387 (sec). Leaf size: 86

DSolve[D[y[x],x] == (1 + x + (3*x^2)/2 - (5*x^3)/2 - x*Cos[x] - 2*x^2*Cos[x] + (15*x^3*Cos[x])/4 + (x^2*Cos[2*x])/2 - (3*x^3*Cos[2*x])/2 + (x^3*Cos[3*x])/4 + x^2*Sin[x] - 2*x*y[x] + (9*x^2*y[x])/2 + 2*x*Cos[x]*y[x] - 6*x^2*Cos[x]*y[x] + (3*x^2*Cos[2*x]*y[x])/2 + y[x]^2 - 3*x*y[x]^2 + 3*x*Cos[x]*y[x]^2 + y[x]^3)/x,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {\frac {3 \cos (x) x-3 x+1}{x}+\frac {3 y(x)}{x}}{\sqrt [3]{29} \sqrt [3]{\frac {1}{x^3}}}}\frac {1}{K[1]^3-\frac {3 K[1]}{29^{2/3}}+1}dK[1]=\frac {1}{9} 29^{2/3} \left (\frac {1}{x^3}\right )^{2/3} x^2 \log (x)+c_1,y(x)\right ] \]