51.4.9 problem 9

Internal problem ID [10366]
Book : First order enumerated odes
Section : section 4. First order odes solved using series method
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 07:22:38 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} x y^{\prime }+2 x y&=\sqrt {x} \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 43

Order:=6; 
ode:=x*diff(y(x),x)+2*x*y(x) = x^(1/2); 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\sqrt {x}\, \left (2-\frac {8}{3} x +\frac {32}{15} x^{2}-\frac {128}{105} x^{3}+\frac {512}{945} x^{4}-\frac {2048}{10395} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 123

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

\[ y(x)\to \left (-\frac {4 x^5}{15}+\frac {2 x^4}{3}-\frac {4 x^3}{3}+2 x^2-2 x+1\right ) \left (\frac {8 x^{11/2}}{165}+\frac {4 x^{9/2}}{27}+\frac {8 x^{7/2}}{21}+\frac {4 x^{5/2}}{5}+\frac {4 x^{3/2}}{3}+2 \sqrt {x}\right )+c_1 \left (-\frac {4 x^5}{15}+\frac {2 x^4}{3}-\frac {4 x^3}{3}+2 x^2-2 x+1\right ) \]

✗ Sympy

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

ValueError : ODE -sqrt(x) + 2*x*y(x) + x*Derivative(y(x), x) does not match hint 1st_power_series