12.14.20 problem 20

Internal problem ID [1961]
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 : 20
Date solved : Monday, January 27, 2025 at 05:39:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-\left (1+3 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: 39

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

\[ y = \frac {c_1 \left (1-\frac {10}{3} x -\frac {35}{18} x^{2}-\frac {14}{81} x^{3}-\frac {7}{3888} x^{4}+\frac {7}{320760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{3}}}+c_2 x \left (1+\frac {2}{7} x +\frac {1}{70} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 65

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

\[ y(x)\to c_1 x \left (\frac {x^2}{70}+\frac {2 x}{7}+1\right )+\frac {c_2 \left (\frac {7 x^5}{320760}-\frac {7 x^4}{3888}-\frac {14 x^3}{81}-\frac {35 x^2}{18}-\frac {10 x}{3}+1\right )}{\sqrt [3]{x}} \]