12.8.17 problem 20
Internal
problem
ID
[1753]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
5
linear
second
order
equations.
Section
5.1
Homogeneous
linear
equations.
Page
203
Problem
number
:
20
Date
solved
:
Tuesday, January 28, 2025 at 02:35:51 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\left (6 x -8\right ) y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 161
dsolve((3*x-1)*diff(y(x),x$2)-(3*x+2)*diff(y(x),x)+(6*x-8)*y(x)=0,y(x), singsol=all)
\[
y = -\frac {6615 \,{\mathrm e}^{-\frac {x \left (i \sqrt {7}-1\right )}{2}} \left (x -\frac {1}{3}\right )^{2} \left (\left (\left (-\frac {23 x}{35}+\frac {296}{735}\right ) \sqrt {7}+i x -\frac {248 i}{105}\right ) c_1 \operatorname {KummerM}\left (\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )-\left (\left (-\frac {x}{5}-\frac {8}{105}\right ) \sqrt {7}+i x -\frac {152 i}{105}\right ) c_2 \operatorname {KummerU}\left (\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )+2 \left (i+\frac {3 \sqrt {7}}{49}\right ) c_1 \operatorname {KummerM}\left (-\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )+\frac {\left (i+\frac {5 \sqrt {7}}{7}\right ) c_2 \operatorname {KummerU}\left (-\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )}{5}\right )}{-225 \sqrt {7}+21 i}
\]
✓ Solution by Mathematica
Time used: 0.110 (sec). Leaf size: 109
DSolve[(3*x-1)*D[y[x],{x,2}]-(3*x+2)*D[y[x],x]+(6*x-8)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to 4 e^{\frac {1}{6} \left (1-i \sqrt {7}\right ) (3 x-1)} (1-3 x)^2 \left (c_1 \operatorname {HypergeometricU}\left (\frac {3}{2}-\frac {5 i}{2 \sqrt {7}},3,\frac {1}{3} i \sqrt {7} (3 x-1)\right )+c_2 L_{-\frac {3}{2}+\frac {5 i}{2 \sqrt {7}}}^2\left (\frac {1}{3} i \sqrt {7} (3 x-1)\right )\right )
\]