12.11.2 problem 12

Internal problem ID [1841]
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 : 12
Date solved : Monday, January 27, 2025 at 05:37:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x^{2}+1\right ) y^{\prime \prime }+3 x^{2} y^{\prime }-2 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: 47

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

\[ y = \left (1+x^{2}-\frac {1}{3} x^{4}-\frac {3}{10} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {1}{4} x^{4}-\frac {4}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 52

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

\[ y(x)\to c_2 \left (-\frac {4 x^5}{15}-\frac {x^4}{4}+\frac {x^3}{3}+x\right )+c_1 \left (-\frac {3 x^5}{10}-\frac {x^4}{3}+x^2+1\right ) \]