12.14.24 problem 24

Internal problem ID [1965]
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 : 24
Date solved : Monday, January 27, 2025 at 05:39:17 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} \left (3+4 x \right ) y^{\prime \prime }+x \left (5+18 x \right ) y^{\prime }-\left (1-12 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(x^2*(3+4*x)*diff(y(x),x$2)+x*(5+18*x)*diff(y(x),x)-(1-12*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{4}/{3}} \left (1-\frac {22}{9} x +\frac {374}{81} x^{2}-\frac {17204}{2187} x^{3}+\frac {249458}{19683} x^{4}-\frac {3492412}{177147} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+2 x -6 x^{2}+12 x^{3}-21 x^{4}+\frac {378}{11} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 80

AsymptoticDSolveValue[x^2*(3+4*x)*D[y[x],{x,2}]+x*(5+18*x)*D[y[x],x]-(1-12*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {3492412 x^5}{177147}+\frac {249458 x^4}{19683}-\frac {17204 x^3}{2187}+\frac {374 x^2}{81}-\frac {22 x}{9}+1\right )+\frac {c_2 \left (\frac {378 x^5}{11}-21 x^4+12 x^3-6 x^2+2 x+1\right )}{x} \]