12.15.30 problem 26

Internal problem ID [2028]
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 : 26
Date solved : Monday, January 27, 2025 at 05:40:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 52

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {3}{4} x^{2}-\frac {7}{12} x^{3}+\frac {91}{192} x^{4}-\frac {637}{1600} x^{5}+\frac {19747}{57600} x^{6}-\frac {17329}{57600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (x -\frac {3}{4} x^{2}+\frac {5}{9} x^{3}-\frac {499}{1152} x^{4}+\frac {16919}{48000} x^{5}-\frac {56861}{192000} x^{6}+\frac {1027717}{4032000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 151

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

\[ y(x)\to c_1 \left (-\frac {17329 x^7}{57600}+\frac {19747 x^6}{57600}-\frac {637 x^5}{1600}+\frac {91 x^4}{192}-\frac {7 x^3}{12}+\frac {3 x^2}{4}-x+1\right )+c_2 \left (\frac {1027717 x^7}{4032000}-\frac {56861 x^6}{192000}+\frac {16919 x^5}{48000}-\frac {499 x^4}{1152}+\frac {5 x^3}{9}-\frac {3 x^2}{4}+\left (-\frac {17329 x^7}{57600}+\frac {19747 x^6}{57600}-\frac {637 x^5}{1600}+\frac {91 x^4}{192}-\frac {7 x^3}{12}+\frac {3 x^2}{4}-x+1\right ) \log (x)+x\right ) \]