46.2.12 problem 12

Internal problem ID [9549]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 06:20:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (x +3\right ) y^{\prime }+7 x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 64

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+(x+3)*diff(y(x),x)+7*x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {7}{15} x^{3}+\frac {7}{120} x^{4}-\frac {1}{150} x^{5}+\frac {11}{160} x^{6}-\frac {197}{15120} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_2 \left (\ln \left (x \right ) \left (2 x^{2}-\frac {14}{15} x^{5}+\frac {7}{60} x^{6}-\frac {1}{75} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2+4 x -3 x^{2}+4 x^{3}-4 x^{4}+\frac {547}{225} x^{5}-\frac {5329}{3600} x^{6}+\frac {7642}{7875} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 96

ode=x*D[y[x],{x,2}]+(x+3)*D[y[x],x]+7*x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {11 x^6}{160}-\frac {x^5}{150}+\frac {7 x^4}{120}-\frac {7 x^3}{15}+1\right )+c_1 \left (\frac {5539 x^6-10432 x^5+14400 x^4-14400 x^3+14400 x^2-14400 x+7200}{7200 x^2}-\frac {1}{120} \left (7 x^4-56 x^3+120\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.314 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(7*x**2*y(x) + x*Derivative(y(x), (x, 2)) + (x + 3)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {197 x^{7}}{15120} + \frac {11 x^{6}}{160} - \frac {x^{5}}{150} + \frac {7 x^{4}}{120} - \frac {7 x^{3}}{15} + 1\right ) + O\left (x^{8}\right ) \]