Internal problem ID [2353]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 38, page 173
Problem number: 16.
ODE order: 1.
ODE degree: 0.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _dAlembert]
\[ \boxed {2 y-3 y^{\prime } x -2 \ln \left (y^{\prime }\right )=4} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 752
dsolve(2*y(x)=3*diff(y(x),x)*x+4+2*ln(diff(y(x),x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {3 x \left (\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{6 x}+\frac {2}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}-\frac {1}{3 x}\right )}{2}+2+\ln \left (\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{6 x}+\frac {2}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}-\frac {1}{3 x}\right ) y \left (x \right ) = \frac {3 x \left (-\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{12 x}-\frac {1}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}-\frac {1}{3 x}-\frac {i \sqrt {3}\, \left (\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{6 x}-\frac {2}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}\right )}{2}\right )}{2}+2+\ln \left (-\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{12 x}-\frac {1}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}-\frac {1}{3 x}-\frac {i \sqrt {3}\, \left (\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{6 x}-\frac {2}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}\right )}{2}\right ) y \left (x \right ) = \frac {3 x \left (-\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{12 x}-\frac {1}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}-\frac {1}{3 x}+\frac {i \sqrt {3}\, \left (\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{6 x}-\frac {2}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}\right )}{2}\right )}{2}+2+\ln \left (-\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{12 x}-\frac {1}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}-\frac {1}{3 x}+\frac {i \sqrt {3}\, \left (\frac {{\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}{6 x}-\frac {2}{3 x {\left (12 \sqrt {3}\, \sqrt {c_{1} \left (27 c_{1} x^{2}-4\right )}\, x +108 c_{1} x^{2}-8\right )}^{\frac {1}{3}}}\right )}{2}\right ) \end{align*}
✓ Solution by Mathematica
Time used: 0.929 (sec). Leaf size: 137
DSolve[2*y[x]==3*y'[x]*x+4+2*Log[y'[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {1}{2} \left (2 W\left (-\frac {3}{2} \sqrt {x^2 e^{2 y(x)-4}}\right )-\log \left (2 W\left (-\frac {3}{2} \sqrt {x^2 e^{2 y(x)-4}}\right )+3\right )+3\right )-y(x)=c_1,y(x)\right ] \text {Solve}\left [\frac {1}{2} \left (2 W\left (\frac {3}{2} \sqrt {x^2 e^{2 y(x)-4}}\right )-\log \left (2 W\left (\frac {3}{2} \sqrt {x^2 e^{2 y(x)-4}}\right )+3\right )+3\right )-y(x)=c_1,y(x)\right ] \end{align*}