28.4.2 problem 2

Internal problem ID [7191]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 04:25:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 44

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

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

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

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

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

✓ Sympy. Time used: 0.234 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
p = symbols("p") 
y = Function("y") 
ode = Eq(p*x*y(x) + x*Derivative(y(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)
 

\[ y{\left (x \right )} = C_{1} \left (\frac {p^{2} x^{4}}{64} - \frac {p x^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]