12.15.4 problem Example 7.6.4 page 372

Internal problem ID [2002]
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 : Example 7.6.4 page 372
Date solved : Monday, January 27, 2025 at 05:40:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (5-x \right ) y^{\prime }+\left (9-4 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 40

Order:=6; 
dsolve(x^2*diff(y(x),x$2)-x*(5-x)*diff(y(x),x)+(9-4*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{3} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+x +\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x -\frac {1}{4} x^{2}+\frac {1}{36} x^{3}-\frac {1}{288} x^{4}+\frac {1}{2400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 62

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-x*(5-x)*D[y[x],x]+(9-4*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 (x+1) x^3+c_2 \left ((x+1) x^3 \log (x)+\left (\frac {x^5}{2400}-\frac {x^4}{288}+\frac {x^3}{36}-\frac {x^2}{4}-3 x\right ) x^3\right ) \]