67.24.22 problem 34.8 b(ii)

Internal problem ID [16989]
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 : 34.8 b(ii)
Date solved : Thursday, October 02, 2025 at 01:41:34 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }+\sqrt {x^{2}+1}\, y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 47

Order:=8; 
ode:=diff(y(x),x)+(x^2+1)^(1/2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-x +\frac {1}{2} x^{2}-\frac {1}{3} x^{3}+\frac {5}{24} x^{4}-\frac {1}{15} x^{5}+\frac {13}{720} x^{6}-\frac {11}{630} x^{7}\right ) y \left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 53

ode=D[y[x],x]+Sqrt[1+x^2]*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {11 x^7}{630}+\frac {13 x^6}{720}-\frac {x^5}{15}+\frac {5 x^4}{24}-\frac {x^3}{3}+\frac {x^2}{2}-x+1\right ) \]

✓ Sympy. Time used: 0.304 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sqrt(x**2 + 1)*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} - C_{1} x + \frac {C_{1} x^{2}}{2} - \frac {C_{1} x^{3}}{3} + \frac {5 C_{1} x^{4}}{24} - \frac {C_{1} x^{5}}{15} + \frac {13 C_{1} x^{6}}{720} - \frac {11 C_{1} x^{7}}{630} + O\left (x^{8}\right ) \]