28.5.11 problem 9.11
Internal
problem
ID
[4598]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
9.
Series
Solutions
of
Differential
Equations.
Problems
at
page
426
Problem
number
:
9.11
Date
solved
:
Monday, January 27, 2025 at 09:25:56 AM
CAS
classification
:
[_Laguerre]
\begin{align*} x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+k y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 154
Order:=6;
dsolve(x*diff(y(x),x$2)+(1-x)*diff(y(x),x)+k*y(x)=0,y(x),type='series',x=0);
\[
y \left (x \right ) = \left (\left (2 k +1\right ) x +\left (\frac {1}{4} k +\frac {1}{4}-\frac {3}{4} k^{2}\right ) x^{2}+\left (-\frac {2}{9} k^{2}+\frac {1}{27} k +\frac {1}{18}+\frac {11}{108} k^{3}\right ) x^{3}+\left (\frac {7}{192} k^{3}-\frac {167}{3456} k^{2}+\frac {1}{192} k +\frac {1}{96}-\frac {25}{3456} k^{4}\right ) x^{4}+\left (\frac {1}{1500} k +\frac {719}{86400} k^{3}+\frac {137}{432000} k^{5}+\frac {1}{600}-\frac {37}{4320} k^{2}-\frac {61}{21600} k^{4}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1-k x +\frac {1}{4} \left (-1+k \right ) k x^{2}-\frac {1}{36} \left (k -2\right ) \left (-1+k \right ) k x^{3}+\frac {1}{576} \left (k -3\right ) \left (k -2\right ) \left (-1+k \right ) k x^{4}-\frac {1}{14400} \left (-4+k \right ) \left (k -3\right ) \left (k -2\right ) \left (-1+k \right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )
\]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 415
AsymptoticDSolveValue[x*D[y[x],{x,2}]+(1-x)*D[y[x],x]+k*y[x]==0,y[x],{x,0,"6"-1}]
\[
y(x)\to c_1 \left (-\frac {(k-4) (k-3) (k-2) (k-1) k x^5}{14400}+\frac {1}{576} (k-3) (k-2) (k-1) k x^4-\frac {1}{36} (k-2) (k-1) k x^3+\frac {1}{4} (k-1) k x^2-k x+1\right )+c_2 \left (\frac {(k-4) (k-3) (k-2) (k-1) x^5}{14400}+\frac {(k-4) (k-3) (k-2) k x^5}{14400}+\frac {(k-4) (k-3) (k-1) k x^5}{14400}+\frac {(k-4) (k-2) (k-1) k x^5}{14400}+\frac {137 (k-4) (k-3) (k-2) (k-1) k x^5}{432000}+\frac {(k-3) (k-2) (k-1) k x^5}{14400}-\frac {1}{576} (k-3) (k-2) (k-1) x^4-\frac {1}{576} (k-3) (k-2) k x^4-\frac {1}{576} (k-3) (k-1) k x^4-\frac {25 (k-3) (k-2) (k-1) k x^4}{3456}-\frac {1}{576} (k-2) (k-1) k x^4+\frac {1}{36} (k-2) (k-1) x^3+\frac {1}{36} (k-2) k x^3+\frac {11}{108} (k-2) (k-1) k x^3+\frac {1}{36} (k-1) k x^3-\frac {1}{4} (k-1) x^2-\frac {3}{4} (k-1) k x^2-\frac {k x^2}{4}+\left (-\frac {(k-4) (k-3) (k-2) (k-1) k x^5}{14400}+\frac {1}{576} (k-3) (k-2) (k-1) k x^4-\frac {1}{36} (k-2) (k-1) k x^3+\frac {1}{4} (k-1) k x^2-k x+1\right ) \log (x)+2 k x+x\right )
\]