Internal problem ID [14825]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Laguerre]
\[ \boxed {x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+y k=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 309
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);
\[ y \left (x \right ) = \left (\left (2 k +1\right ) x +\left (\frac {1}{4} k +\frac {1}{4}-\frac {3}{4} k^{2}\right ) x^{2}+\left (-\frac {2}{9} k^{2}+\frac {1}{27} k +\frac {1}{18}+\frac {11}{108} k^{3}\right ) x^{3}+\left (\frac {7}{192} k^{3}-\frac {167}{3456} k^{2}+\frac {1}{192} k +\frac {1}{96}-\frac {25}{3456} k^{4}\right ) x^{4}+\left (\frac {1}{1500} k -\frac {61}{21600} k^{4}+\frac {1}{600}+\frac {719}{86400} k^{3}-\frac {37}{4320} k^{2}+\frac {137}{432000} k^{5}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1-k x +\frac {1}{4} \left (-1+k \right ) k x^{2}-\frac {1}{36} \left (-2+k \right ) \left (-1+k \right ) k x^{3}+\frac {1}{576} \left (k -3\right ) \left (-2+k \right ) \left (-1+k \right ) k x^{4}-\frac {1}{14400} \left (-4+k \right ) \left (k -3\right ) \left (-2+k \right ) \left (-1+k \right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_{1} +c_{2} \ln \left (x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 415
AsymptoticDSolveValue[x*y''[x]+(1-x)*y'[x]+k*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {(k-4) (k-3) (k-2) (k-1) k x^5}{14400}+\frac {1}{576} (k-3) (k-2) (k-1) k x^4-\frac {1}{36} (k-2) (k-1) k x^3+\frac {1}{4} (k-1) k x^2-k x+1\right )+c_2 \left (\frac {(k-4) (k-3) (k-2) (k-1) x^5}{14400}+\frac {(k-4) (k-3) (k-2) k x^5}{14400}+\frac {(k-4) (k-3) (k-1) k x^5}{14400}+\frac {(k-4) (k-2) (k-1) k x^5}{14400}+\frac {137 (k-4) (k-3) (k-2) (k-1) k x^5}{432000}+\frac {(k-3) (k-2) (k-1) k x^5}{14400}-\frac {1}{576} (k-3) (k-2) (k-1) x^4-\frac {1}{576} (k-3) (k-2) k x^4-\frac {1}{576} (k-3) (k-1) k x^4-\frac {25 (k-3) (k-2) (k-1) k x^4}{3456}-\frac {1}{576} (k-2) (k-1) k x^4+\frac {1}{36} (k-2) (k-1) x^3+\frac {1}{36} (k-2) k x^3+\frac {11}{108} (k-2) (k-1) k x^3+\frac {1}{36} (k-1) k x^3-\frac {1}{4} (k-1) x^2-\frac {3}{4} (k-1) k x^2-\frac {k x^2}{4}+\left (-\frac {(k-4) (k-3) (k-2) (k-1) k x^5}{14400}+\frac {1}{576} (k-3) (k-2) (k-1) k x^4-\frac {1}{36} (k-2) (k-1) k x^3+\frac {1}{4} (k-1) k x^2-k x+1\right ) \log (x)+2 k x+x\right ) \]