68.17.24 problem 24

Internal problem ID [17842]
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 : Thursday, October 02, 2025 at 02:28:48 PM
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*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 154

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+(1-x)*diff(y(x),x)+k*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (\left (2 k +1\right ) x +\left (\frac {1}{4} k +\frac {1}{4}-\frac {3}{4} k^{2}\right ) x^{2}+\left (-\frac {2}{9} k^{2}+\frac {1}{27} k +\frac {1}{18}+\frac {11}{108} k^{3}\right ) x^{3}+\left (\frac {7}{192} k^{3}-\frac {167}{3456} k^{2}+\frac {1}{192} k +\frac {1}{96}-\frac {25}{3456} k^{4}\right ) x^{4}+\left (\frac {719}{86400} k^{3}-\frac {61}{21600} k^{4}+\frac {137}{432000} k^{5}+\frac {1}{600}-\frac {37}{4320} k^{2}+\frac {1}{1500} k \right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-k x +\frac {1}{4} \left (-1+k \right ) k x^{2}-\frac {1}{36} \left (k -2\right ) \left (-1+k \right ) k x^{3}+\frac {1}{576} \left (k -3\right ) \left (k -2\right ) \left (-1+k \right ) k x^{4}-\frac {1}{14400} \left (k -4\right ) \left (k -3\right ) \left (k -2\right ) \left (-1+k \right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 415

ode=x*D[y[x],{x,2}]+(1-x)*D[y[x],x]+k*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {(k-4) (k-3) (k-2) (k-1) k x^5}{14400}+\frac {1}{576} (k-3) (k-2) (k-1) k x^4-\frac {1}{36} (k-2) (k-1) k x^3+\frac {1}{4} (k-1) k x^2-k x+1\right )+c_2 \left (\frac {(k-4) (k-3) (k-2) (k-1) x^5}{14400}+\frac {(k-4) (k-3) (k-2) k x^5}{14400}+\frac {(k-4) (k-3) (k-1) k x^5}{14400}+\frac {(k-4) (k-2) (k-1) k x^5}{14400}+\frac {137 (k-4) (k-3) (k-2) (k-1) k x^5}{432000}+\frac {(k-3) (k-2) (k-1) k x^5}{14400}-\frac {1}{576} (k-3) (k-2) (k-1) x^4-\frac {1}{576} (k-3) (k-2) k x^4-\frac {1}{576} (k-3) (k-1) k x^4-\frac {25 (k-3) (k-2) (k-1) k x^4}{3456}-\frac {1}{576} (k-2) (k-1) k x^4+\frac {1}{36} (k-2) (k-1) x^3+\frac {1}{36} (k-2) k x^3+\frac {11}{108} (k-2) (k-1) k x^3+\frac {1}{36} (k-1) k x^3-\frac {1}{4} (k-1) x^2-\frac {3}{4} (k-1) k x^2-\frac {k x^2}{4}+\left (-\frac {(k-4) (k-3) (k-2) (k-1) k x^5}{14400}+\frac {1}{576} (k-3) (k-2) (k-1) k x^4-\frac {1}{36} (k-2) (k-1) k x^3+\frac {1}{4} (k-1) k x^2-k x+1\right ) \log (x)+2 k x+x\right ) \]

✓ Sympy. Time used: 0.350 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(k*y(x) + x*Derivative(y(x), (x, 2)) + (1 - x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {k x^{5} \left (k - 4\right ) \left (k - 3\right ) \left (k - 2\right ) \left (k - 1\right )}{14400} + \frac {k x^{4} \left (k - 3\right ) \left (k - 2\right ) \left (k - 1\right )}{576} - \frac {k x^{3} \left (k - 2\right ) \left (k - 1\right )}{36} + \frac {k x^{2} \left (k - 1\right )}{4} - k x + 1\right ) + O\left (x^{6}\right ) \]