85.79.2 problem 7

Internal problem ID [22991]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. C Exercises at page 342
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:17:20 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+y^{\prime }-i x y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 44

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{4} i \,x^{2}+\frac {1}{64} i^{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {i}{4} x^{2}-\frac {3}{128} i^{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 72

ode=x*D[y[x],{x,2}]+D[y[x],x]-i*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {i^2 x^4}{64}+\frac {i x^2}{4}+1\right )+c_2 \left (-\frac {3}{128} i^2 x^4+\left (\frac {i^2 x^4}{64}+\frac {i x^2}{4}+1\right ) \log (x)-\frac {i x^2}{4}\right ) \]

✗ Sympy

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

ValueError : ODE -i*x*y(x) + x*Derivative(v(x), (x, 2)) + Derivative(y(x), x) does not match hint 2n