12.16.19 problem 15

Internal problem ID [2081]
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 III. Exercises 7.7. Page 389
Problem number : 15
Date solved : Monday, January 27, 2025 at 05:41:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (4-x \right ) y^{\prime }-\left (7+5 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: 42

Order:=6; 
dsolve(4*x^2*(1+2*x)*diff(y(x),x$2)-2*x*(4-x)*diff(y(x),x)-(7+5*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {18}{5} x +\frac {39}{4} x^{2}-\frac {663}{28} x^{3}+\frac {13923}{256} x^{4}-\frac {7735}{64} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-144-\frac {405}{8} x^{4}+\frac {729}{4} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 67

AsymptoticDSolveValue[4*x^2*(1+2*x)*D[y[x],{x,2}]-2*x*(4-x)*D[y[x],x]-(7+5*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{\sqrt {x}}-\frac {35 x^{7/2}}{128}\right )+c_2 \left (\frac {13923 x^{15/2}}{256}-\frac {663 x^{13/2}}{28}+\frac {39 x^{11/2}}{4}-\frac {18 x^{9/2}}{5}+x^{7/2}\right ) \]