12.16.27 problem 23

Internal problem ID [2089]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 23
Date solved : Monday, January 27, 2025 at 05:42:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+7\right ) y^{\prime }+12 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 46

Order:=6; 
dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(7-2*x^2)*diff(y(x),x)+12*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{2} \left (c_1 \,x^{4} \left (1-\frac {7}{2} x^{2}+\frac {63}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (1080 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-216 x^{2}+2106 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 57

AsymptoticDSolveValue[x^2*(1+x^2)*D[y[x],{x,2}]-x*(7-2*x^2)*D[y[x],x]+12*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {63 x^{10}}{8}-\frac {7 x^8}{2}+x^6\right )+c_1 \left (-\frac {15}{2} x^6 \log (x)-\frac {1}{4} \left (31 x^4-6 x^2-4\right ) x^2\right ) \]