12.11.3 problem 13

Internal problem ID [1842]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 13
Date solved : Monday, January 27, 2025 at 05:37:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }+\left (2-3 x \right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 59

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

\[ y = \left (1-2 x^{2}+\frac {4}{3} x^{3}-\frac {1}{3} x^{4}-\frac {1}{3} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {1}{2} x^{3}-\frac {1}{12} x^{4}-\frac {17}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 66

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

\[ y(x)\to c_1 \left (-\frac {x^5}{3}-\frac {x^4}{3}+\frac {4 x^3}{3}-2 x^2+1\right )+c_2 \left (-\frac {17 x^5}{120}-\frac {x^4}{12}+\frac {x^3}{2}-x^2+x\right ) \]