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`]]
\[ \boxed {y^{\prime }+\frac {4 x +3 y}{2 x +y}=0} \]
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 1228
dsolve(diff(y(x),x) = - (4*x+3*y(x))/(2*x+y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\frac {\left (-4 c_{1} x^{3}+\left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}\right )^{2}}{4 \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}} c_{1}}-x^{3}}{x^{2}} \\ y \left (x \right ) &= \frac {-3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} +\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{6} c_{1}^{2}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= \frac {-3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} +\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{6} c_{1}^{2}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= -\frac {\frac {\left (4 \sqrt {3}\, c_{1} x^{3}+\sqrt {3}\, \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}+4 i c_{1} x^{3}-i \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}\right )^{2}}{16 \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}} c_{1}}+x^{3}}{x^{2}} \\ y \left (x \right ) &= -\frac {\frac {\left (4 \sqrt {3}\, c_{1} x^{3}+\sqrt {3}\, \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}-4 i c_{1} x^{3}+i \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}\right )^{2}}{16 \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}} c_{1}}+x^{3}}{x^{2}} \\ y \left (x \right ) &= -\frac {2 \left (\frac {3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1}}{2}-\frac {\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (i \sqrt {3}-1\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+x^{6} \left (1+i \sqrt {3}\right ) c_{1}^{2}\right )}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= \frac {-3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} -\frac {\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (1+i \sqrt {3}\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+2 x^{6} \left (i \sqrt {3}-1\right ) c_{1}^{2}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= -\frac {2 \left (\frac {3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1}}{2}-\frac {\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (i \sqrt {3}-1\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+x^{6} \left (1+i \sqrt {3}\right ) c_{1}^{2}\right )}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= \frac {-3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} -\frac {\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (1+i \sqrt {3}\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+2 x^{6} \left (i \sqrt {3}-1\right ) c_{1}^{2}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 20.375 (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*}