74.17.24 problem 24
Internal
problem
ID
[16555]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
4.
Higher
Order
Equations.
Exercises
4.9,
page
215
Problem
number
:
24
Date
solved
:
Tuesday, January 28, 2025 at 09:10:43 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: 0.051 (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);
\[
y = \left (\left (1+2 k \right ) x +\left (\frac {1}{4} k +\frac {1}{4}-\frac {3}{4} k^{2}\right ) x^{2}+\left (-\frac {2}{9} k^{2}+\frac {1}{27} k +\frac {1}{18}+\frac {11}{108} k^{3}\right ) x^{3}+\left (\frac {7}{192} k^{3}-\frac {167}{3456} k^{2}+\frac {1}{192} k +\frac {1}{96}-\frac {25}{3456} k^{4}\right ) x^{4}+\left (\frac {1}{1500} k -\frac {61}{21600} k^{4}-\frac {37}{4320} k^{2}+\frac {1}{600}+\frac {719}{86400} k^{3}+\frac {137}{432000} k^{5}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1-k x +\frac {1}{4} \left (k -1\right ) k x^{2}-\frac {1}{36} \left (-2+k \right ) \left (k -1\right ) k x^{3}+\frac {1}{576} \left (k -3\right ) \left (-2+k \right ) \left (k -1\right ) k x^{4}-\frac {1}{14400} \left (k -4\right ) \left (k -3\right ) \left (-2+k \right ) \left (k -1\right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (\ln \left (x \right ) c_{2} +c_{1} \right )
\]
✓ 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}]
\[
y(x)\to c_1 \left (-\frac {(k-4) (k-3) (k-2) (k-1) k x^5}{14400}+\frac {1}{576} (k-3) (k-2) (k-1) k x^4-\frac {1}{36} (k-2) (k-1) k x^3+\frac {1}{4} (k-1) k x^2-k x+1\right )+c_2 \left (\frac {(k-4) (k-3) (k-2) (k-1) x^5}{14400}+\frac {(k-4) (k-3) (k-2) k x^5}{14400}+\frac {(k-4) (k-3) (k-1) k x^5}{14400}+\frac {(k-4) (k-2) (k-1) k x^5}{14400}+\frac {137 (k-4) (k-3) (k-2) (k-1) k x^5}{432000}+\frac {(k-3) (k-2) (k-1) k x^5}{14400}-\frac {1}{576} (k-3) (k-2) (k-1) x^4-\frac {1}{576} (k-3) (k-2) k x^4-\frac {1}{576} (k-3) (k-1) k x^4-\frac {25 (k-3) (k-2) (k-1) k x^4}{3456}-\frac {1}{576} (k-2) (k-1) k x^4+\frac {1}{36} (k-2) (k-1) x^3+\frac {1}{36} (k-2) k x^3+\frac {11}{108} (k-2) (k-1) k x^3+\frac {1}{36} (k-1) k x^3-\frac {1}{4} (k-1) x^2-\frac {3}{4} (k-1) k x^2-\frac {k x^2}{4}+\left (-\frac {(k-4) (k-3) (k-2) (k-1) k x^5}{14400}+\frac {1}{576} (k-3) (k-2) (k-1) k x^4-\frac {1}{36} (k-2) (k-1) k x^3+\frac {1}{4} (k-1) k x^2-k x+1\right ) \log (x)+2 k x+x\right )
\]