54.9.18 problem 19
Internal
problem
ID
[8688]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
18.
Power
series
solutions.
Miscellaneous
Exercises.
page
394
Problem
number
:
19
Date
solved
:
Wednesday, March 05, 2025 at 06:12:28 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 4 x^{2} y^{\prime \prime }+3 x^{2} y^{\prime }+\left (1+3 x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 56
Order:=8;
ode:=4*x^2*diff(diff(y(x),x),x)+3*x^2*diff(y(x),x)+(3*x+1)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {9}{8} x +\frac {135}{256} x^{2}-\frac {315}{2048} x^{3}+\frac {8505}{262144} x^{4}-\frac {56133}{10485760} x^{5}+\frac {243243}{335544320} x^{6}-\frac {312741}{3758096384} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{2} x -\frac {261}{256} x^{2}+\frac {729}{2048} x^{3}-\frac {44091}{524288} x^{4}+\frac {63099}{4194304} x^{5}-\frac {1454463}{671088640} x^{6}+\frac {1403811}{5368709120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \sqrt {x}
\]
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 176
ode=4*x^2*D[y[x],{x,2}]+3*x^2*D[y[x],x]+(1+3*x)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \sqrt {x} \left (-\frac {312741 x^7}{3758096384}+\frac {243243 x^6}{335544320}-\frac {56133 x^5}{10485760}+\frac {8505 x^4}{262144}-\frac {315 x^3}{2048}+\frac {135 x^2}{256}-\frac {9 x}{8}+1\right )+c_2 \left (\sqrt {x} \left (\frac {1403811 x^7}{5368709120}-\frac {1454463 x^6}{671088640}+\frac {63099 x^5}{4194304}-\frac {44091 x^4}{524288}+\frac {729 x^3}{2048}-\frac {261 x^2}{256}+\frac {3 x}{2}\right )+\sqrt {x} \left (-\frac {312741 x^7}{3758096384}+\frac {243243 x^6}{335544320}-\frac {56133 x^5}{10485760}+\frac {8505 x^4}{262144}-\frac {315 x^3}{2048}+\frac {135 x^2}{256}-\frac {9 x}{8}+1\right ) \log (x)\right )
\]
✓ Sympy. Time used: 0.948 (sec). Leaf size: 53
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(3*x**2*Derivative(y(x), x) + 4*x**2*Derivative(y(x), (x, 2)) + (3*x + 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {243243 x^{6}}{335544320} - \frac {56133 x^{5}}{10485760} + \frac {8505 x^{4}}{262144} - \frac {315 x^{3}}{2048} + \frac {135 x^{2}}{256} - \frac {9 x}{8} + 1\right ) + O\left (x^{8}\right )
\]