Internal problem ID [15141]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Section 12. Miscellaneous problems. Exercises page 93
Problem number: 279.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\[ \boxed {5 y x -4 y^{2}+\left (y^{2}-8 y x +\frac {5 x^{2}}{2}\right ) y^{\prime }=6 x^{2}} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 443
dsolve((5*x*y(x)-4*y(x)^2-6*x^2)+(y(x)^2-8*x*y(x)+25/10*x^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y &= \frac {\frac {\left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {27 x^{2} c_{1}^{2}}{\left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+4 c_{1} x}{c_{1}} \\ y &= \frac {54 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}} \sqrt {3}-54 x^{2} c_{1}^{2}+16 c_{1} x \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}-\left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}}}{4 \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}} c_{1}} \\ y &= -\frac {54 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}} \sqrt {3}+54 x^{2} c_{1}^{2}-16 c_{1} x \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}+\left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}}}{4 \left (416 x^{3} c_{1}^{3}+2+2 \sqrt {3898 c_{1}^{6} x^{6}+416 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 29.95 (sec). Leaf size: 741
DSolve[(5*x*y[x]-4*y[x]^2-6*x^2)+(y[x]^2-8*x*y[x]+25/10*x^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2^{2/3}}+\frac {27 x^2}{\sqrt [3]{2} \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+4 x \\ y(x)\to -\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2\ 2^{2/3}}-\frac {27 \left (1+i \sqrt {3}\right ) x^2}{2 \sqrt [3]{2} \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+4 x \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2\ 2^{2/3}}-\frac {27 \left (1-i \sqrt {3}\right ) x^2}{2 \sqrt [3]{2} \sqrt [3]{208 x^3+\sqrt {3898 x^6+416 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}+4 x \\ y(x)\to \frac {27\ 2^{2/3} x^2+8 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3} x+\sqrt [3]{2} \left (\sqrt {3898} \sqrt {x^6}+208 x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3}} \\ y(x)\to \frac {27 i 2^{2/3} \sqrt {3} x^2-27\ 2^{2/3} x^2+16 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3} x-i \sqrt [3]{2} \sqrt {3} \left (\sqrt {3898} \sqrt {x^6}+208 x^3\right )^{2/3}-\sqrt [3]{2} \left (\sqrt {3898} \sqrt {x^6}+208 x^3\right )^{2/3}}{4 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3}} \\ y(x)\to \frac {\left (\sqrt {3898} \sqrt {x^6}+208 x^3\right )^{2/3} \text {Root}\left [\text {$\#$1}^3-16\&,3\right ]-54 \sqrt [3]{-1} 2^{2/3} x^2+16 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3} x}{4 \sqrt [3]{\sqrt {3898} \sqrt {x^6}+208 x^3}} \\ \end{align*}