46.2.10 problem 10

Internal problem ID [9547]
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 : 10
Date solved : Tuesday, September 30, 2025 at 06:20:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}-2 x^{2}+3 x \right )^{2} y^{\prime \prime }+x \left (x -3\right )^{2} y^{\prime }-\left (x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 56

Order:=8; 
ode:=(x^3-2*x^2+3*x)^2*diff(diff(y(x),x),x)+x*(x-3)^2*diff(y(x),x)-(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{2}/{3}} \left (1+\frac {1}{45} x +\frac {149}{3240} x^{2}+\frac {2701}{192456} x^{3}+\frac {236933}{121247280} x^{4}-\frac {67092967}{92754169200} x^{5}-\frac {30839263691}{50087251368000} x^{6}-\frac {14846109458423}{72576427232232000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_1 \left (1+\frac {13}{9} x -\frac {5}{162} x^{2}+\frac {1591}{30618} x^{3}+\frac {106583}{5511240} x^{4}+\frac {7435523}{3224075400} x^{5}-\frac {70024699}{43525017900} x^{6}-\frac {2917066898}{2604972321315} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 118

ode=(x^3-2*x^2+3*x)^2*D[y[x],{x,2}]+x*(x-3)^2*D[y[x],x]-(x+1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {14846109458423 x^7}{72576427232232000}-\frac {30839263691 x^6}{50087251368000}-\frac {67092967 x^5}{92754169200}+\frac {236933 x^4}{121247280}+\frac {2701 x^3}{192456}+\frac {149 x^2}{3240}+\frac {x}{45}+1\right )+\frac {c_2 \left (-\frac {2917066898 x^7}{2604972321315}-\frac {70024699 x^6}{43525017900}+\frac {7435523 x^5}{3224075400}+\frac {106583 x^4}{5511240}+\frac {1591 x^3}{30618}-\frac {5 x^2}{162}+\frac {13 x}{9}+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 0.738 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x - 3)**2*Derivative(y(x), x) - (x + 1)*y(x) + (x**3 - 2*x**2 + 3*x)**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \sqrt [3]{x} + \frac {C_{1}}{\sqrt [3]{x}} + O\left (x^{8}\right ) \]