73.25.27 problem 35.4 (m)

Internal problem ID [15735]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.4 (m)
Date solved : Tuesday, January 28, 2025 at 08:06:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 28

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

\[ y = \left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1-\frac {1}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 39

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

\[ y(x)\to c_1 \left (1-\frac {x^3}{3}\right )+c_2 \left (\frac {x^3}{3}+\left (1-\frac {x^3}{3}\right ) \log (x)\right ) \]