12.14.41 problem 43

Internal problem ID [1982]
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 : 43
Date solved : Monday, January 27, 2025 at 05:39:39 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x^{2}+1\right ) y^{\prime \prime }+\left (7 x^{2}+4\right ) y^{\prime }+8 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.731 (sec). Leaf size: 32

Order:=6; 
dsolve(x*(1+x^2)*diff(y(x),x$2)+(4+7*x^2)*diff(y(x),x)+8*x*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {4}{5} x^{2}+\frac {24}{35} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12-6 x^{2}+\frac {9}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 42

AsymptoticDSolveValue[x*(1+x^2)*D[y[x],{x,2}]+(4+7*x^2)*D[y[x],x]+8*x*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {3 x}{8}-\frac {1}{2 x}\right )+c_2 \left (\frac {24 x^4}{35}-\frac {4 x^2}{5}+1\right ) \]