73.25.24 problem 35.4 (j)

Internal problem ID [15732]
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 (j)
Date solved : Tuesday, January 28, 2025 at 08:06:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-3 x^{3}+3 x^{2}\right ) y^{\prime \prime }-\left (5 x^{2}+4 x \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 47

Order:=6; 
dsolve((3*x^2-3*x^3)*diff(y(x),x$2)-(4*x+5*x^2)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{{1}/{3}} \left (1-\frac {1}{2} x -2 x^{2}-\frac {7}{2} x^{3}-5 x^{4}-\frac {13}{2} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 74

AsymptoticDSolveValue[(3*x^2-3*x^3)*D[y[x],{x,2}]-(4*x+5*x^2)*D[y[x],x]+2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (6 x^5+5 x^4+4 x^3+3 x^2+2 x+1\right ) x^2+c_2 \left (-\frac {13 x^5}{2}-5 x^4-\frac {7 x^3}{2}-2 x^2-\frac {x}{2}+1\right ) \sqrt [3]{x} \]