15.20.16 problem 16
Internal
problem
ID
[3324]
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
Date
solved
:
Monday, January 27, 2025 at 07:34:04 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} 2 y&=3 x y^{\prime }+4+2 \ln \left (y^{\prime }\right ) \end{align*}
✓ Solution by Maple
Time used: 0.010 (sec). Leaf size: 766
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 )^{{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 )^{{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 )^{{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 )^{{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 )^{{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 )^{{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 )^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {-8 \ln \left (\frac {i \left (4-\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{{2}/{3}}\right ) \sqrt {3}-{\left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{{1}/{3}}+2\right )}^{2}}{x \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{{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 )^{{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 )^{{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 )^{{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 )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {8 \ln \left (\frac {i \left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{{2}/{3}}-4\right ) \sqrt {3}-{\left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{{1}/{3}}+2\right )}^{2}}{x \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{{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 )^{{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 )^{{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 )^{{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 )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.908 (sec). Leaf size: 137
DSolve[2*y[x]==3*D[y[x],x]*x+4+2*Log[D[y[x],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*}