Internal problem ID [588]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Miscellaneous problems, end of chapter 2. Page 133
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {\frac {-4+6 y x +2 y^{2}}{3 x^{2}+4 y x +3 y^{2}}+y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 561
dsolve((-4+6*x*y(x)+2*y(x)^2)/(3*x^2+4*x*y(x)+3*y(x)^2)+diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {10 x^{2}}{3 \left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}-\frac {2 x}{3} \\ y \relax (x ) = -\frac {\left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}{12}+\frac {5 x^{2}}{3 \left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}-\frac {2 x}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {10 x^{2}}{3 \left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}{12}+\frac {5 x^{2}}{3 \left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}-\frac {2 x}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}{6}+\frac {10 x^{2}}{3 \left (152 x^{3}-108 c_{1}+432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 x c_{1}+1296 x^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.867 (sec). Leaf size: 365
DSolve[(-4+6*x*y[x]+2*y[x]^2)/(3*x^2+4*x*y[x]+3*y[x]^2)+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}-\frac {10 \sqrt [3]{2} x^2}{\sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}}-4 x\right ) \\ y(x)\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}+\frac {20 \sqrt [3]{-2} x^2}{\sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}}-8 x\right ) \\ y(x)\to \frac {1}{12} \left (\frac {x^2 \text {Root}\left [\text {$\#$1}^3+16000\&,2\right ]}{\sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}}-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{38 x^3+\sqrt {500 x^6+\left (38 x^3+108 x+27 c_1\right ){}^2}+108 x+27 c_1}-8 x\right ) \\ \end{align*}