12.15.20 problem 16

Internal problem ID [2018]
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 : 16
Date solved : Monday, January 27, 2025 at 05:40:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 25 x^{2} y^{\prime \prime }+x \left (15+x \right ) y^{\prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 48

Order:=6; 
dsolve(25*x^2*diff(y(x),x$2)+x*(15+x)*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{{1}/{5}} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {6}{125} x +\frac {33}{31250} x^{2}-\frac {88}{5859375} x^{3}+\frac {77}{488281250} x^{4}-\frac {1001}{762939453125} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {7}{125} x -\frac {113}{62500} x^{2}+\frac {1091}{35156250} x^{3}-\frac {1721}{4687500000} x^{4}+\frac {609221}{183105468750000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 134

AsymptoticDSolveValue[25*x^2*D[y[x],{x,2}]+x*(15+x)*D[y[x],x]+(1+x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt [5]{x} \left (-\frac {1001 x^5}{762939453125}+\frac {77 x^4}{488281250}-\frac {88 x^3}{5859375}+\frac {33 x^2}{31250}-\frac {6 x}{125}+1\right )+c_2 \left (\sqrt [5]{x} \left (\frac {609221 x^5}{183105468750000}-\frac {1721 x^4}{4687500000}+\frac {1091 x^3}{35156250}-\frac {113 x^2}{62500}+\frac {7 x}{125}\right )+\sqrt [5]{x} \left (-\frac {1001 x^5}{762939453125}+\frac {77 x^4}{488281250}-\frac {88 x^3}{5859375}+\frac {33 x^2}{31250}-\frac {6 x}{125}+1\right ) \log (x)\right ) \]