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]
\[ \boxed {2 y-3 x y^{\prime }-2 \ln \left (y^{\prime }\right )=4} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 827
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 {\ln \left (\frac {\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {2}{3}}-2 \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+4}{x \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+\left (-\ln \left (2\right )-\ln \left (3\right )+\frac {3}{2}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+\frac {\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {2}{3}}}{4}+1}{\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {-8 \ln \left (-\frac {\left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+2\right ) \left (2+i \left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}-2\right ) \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}\right )}{x \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {2}{3}}+\left (16 \ln \left (2\right )+8 \ln \left (3\right )-12\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}-4 i \sqrt {3}+4}{8 \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {8 \ln \left (\frac {\left (-2+i \left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}-2\right ) \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}\right ) \left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+2\right )}{x \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {2}{3}}+\left (-16 \ln \left (2\right )-8 \ln \left (3\right )+12\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}-4 i \sqrt {3}-4}{8 \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ \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*}