12.14.28 problem 30

Internal problem ID [1969]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 30
Date solved : Monday, January 27, 2025 at 05:39:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+11 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: 0.011 (sec). Leaf size: 46

Order:=6; 
dsolve(2*x^2*(2+3*x)*diff(y(x),x$2)+x*(4+11*x)*diff(y(x),x)-(1-x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 x \left (1-\frac {5}{8} x +\frac {55}{96} x^{2}-\frac {935}{1536} x^{3}+\frac {4301}{6144} x^{4}-\frac {124729}{147456} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {5}{4} x +\frac {25}{32} x^{2}-\frac {275}{384} x^{3}+\frac {4675}{6144} x^{4}-\frac {21505}{24576} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 94

AsymptoticDSolveValue[2*x^2*(2+3*x)*D[y[x],{x,2}]+x*(4+11*x)*D[y[x],x]-(1-x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {935 x^{7/2}}{6144}+\frac {55 x^{5/2}}{384}-\frac {5 x^{3/2}}{32}+\frac {\sqrt {x}}{4}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {4301 x^{9/2}}{6144}-\frac {935 x^{7/2}}{1536}+\frac {55 x^{5/2}}{96}-\frac {5 x^{3/2}}{8}+\sqrt {x}\right ) \]