12.15.11 problem 7

Internal problem ID [2009]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 7
Date solved : Monday, January 27, 2025 at 05:40:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 52

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +\frac {7}{4} x^{2}-\frac {7}{9} x^{3}+\frac {77}{576} x^{4}+\frac {217}{7200} x^{5}-\frac {8813}{518400} x^{6}+\frac {143}{453600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (3 x -\frac {15}{4} x^{2}+\frac {239}{108} x^{3}-\frac {2021}{3456} x^{4}-\frac {1241}{54000} x^{5}+\frac {93859}{1728000} x^{6}-\frac {311177}{42336000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 153

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*(1+x+x^2)*D[y[x],x]+x*(2-x)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {143 x^7}{453600}-\frac {8813 x^6}{518400}+\frac {217 x^5}{7200}+\frac {77 x^4}{576}-\frac {7 x^3}{9}+\frac {7 x^2}{4}-2 x+1\right )+c_2 \left (-\frac {311177 x^7}{42336000}+\frac {93859 x^6}{1728000}-\frac {1241 x^5}{54000}-\frac {2021 x^4}{3456}+\frac {239 x^3}{108}-\frac {15 x^2}{4}+\left (\frac {143 x^7}{453600}-\frac {8813 x^6}{518400}+\frac {217 x^5}{7200}+\frac {77 x^4}{576}-\frac {7 x^3}{9}+\frac {7 x^2}{4}-2 x+1\right ) \log (x)+3 x\right ) \]