Internal problem ID [511]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.2. Page 48
Problem number: 34.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {4 x +3 y}{2 x +y}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 1282
dsolve(diff(y(x),x) = - (4*x+3*y(x))/(2*x+y(x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\frac {\left (4 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}-\frac {16 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ y \relax (x ) = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} \left (4 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}-\frac {16 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ y \relax (x ) = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} \left (4 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}-\frac {16 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ y \relax (x ) = \frac {\frac {\left (-2 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ y \relax (x ) = \frac {\frac {\left (-2 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ y \relax (x ) = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} \left (-2 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ y \relax (x ) = \frac {\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{6} \left (-2 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ y \relax (x ) = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} \left (-2 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}-4 i \sqrt {3}\, \left (\frac {\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ y \relax (x ) = \frac {\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{6} \left (-2 \left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+\frac {8 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}+4 i \sqrt {3}\, \left (\frac {\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3} c_{1}}{\left (4 x^{3} c_{1}+4 \sqrt {4 x^{9} c_{1}^{3}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}\right )\right )^{2}}{64 c_{1}}-x^{3}}{x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 20.9 (sec). Leaf size: 484
DSolve[y'[x] == - (4*x+3*y[x])/(2*x+y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2 x^3+\sqrt {4 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{2 x^3+\sqrt {4 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-3 x \\ y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^3+\sqrt {4 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{2 x^3+\sqrt {4 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-3 x \\ y(x)\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^3+\sqrt {4 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{2^{2/3} \sqrt [3]{2 x^3+\sqrt {4 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-3 x \\ y(x)\to \sqrt [3]{x^3}+\frac {\left (x^3\right )^{2/3}}{x}-3 x \\ y(x)\to \frac {1}{2} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3}+\frac {\left (-1-i \sqrt {3}\right ) \left (x^3\right )^{2/3}}{x}-6 x\right ) \\ y(x)\to \frac {1}{2} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^3}+\frac {i \left (\sqrt {3}+i\right ) \left (x^3\right )^{2/3}}{x}-6 x\right ) \\ \end{align*}