67.24.28 problem 34.9 b(iv)

Internal problem ID [16995]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.9 b(iv)
Date solved : Thursday, October 02, 2025 at 01:41:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 62

Order:=5; 
ode:=x^(1/2)*diff(diff(y(x),x),x)+diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{12}-\frac {\left (x -1\right )^{4}}{96}\right ) y \left (1\right )+\left (x -1-\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{3}}{12}-\frac {3 \left (x -1\right )^{4}}{32}\right ) y^{\prime }\left (1\right )+O\left (x^{5}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 69

ode=Sqrt[x]*D[y[x],{x,2}]+D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,4}]
 

\[ y(x)\to c_1 \left (-\frac {1}{96} (x-1)^4+\frac {1}{12} (x-1)^3-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (-\frac {3}{32} (x-1)^4+\frac {1}{12} (x-1)^3-\frac {1}{2} (x-1)^2+x-1\right ) \]

✓ Sympy. Time used: 0.373 (sec). Leaf size: 88

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sqrt(x)*Derivative(y(x), (x, 2)) + x*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=5)
 

\[ y{\left (x \right )} = C_{2} \left (x + \frac {\left (x - 1\right )^{3}}{6 \left (x + 1\right )} - \frac {\left (x - 1\right )^{3}}{6 \sqrt {x + 1}} - \frac {\left (x - 1\right )^{2}}{2 \sqrt {x + 1}} - 1\right ) + C_{1} \left (\frac {\left (x - 1\right )^{3}}{6 \left (x + 1\right )} - \frac {\left (x - 1\right )^{3}}{6 \sqrt {x + 1}} - \frac {\left (x - 1\right )^{2}}{2 \sqrt {x + 1}} + 1\right ) + O\left (x^{5}\right ) \]