56.4.22 problem 22

Internal problem ID [8911]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 22
Date solved : Monday, January 27, 2025 at 05:19:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (3 x^{2}+2 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: 0.016 (sec). Leaf size: 45

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

\[ y = c_{1} x \left (1-\frac {3}{4} x +\frac {9}{20} x^{2}-\frac {9}{40} x^{3}+\frac {27}{280} x^{4}-\frac {81}{2240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-36 x +54 x^{2}-54 x^{3}+\frac {81}{2} x^{4}-\frac {243}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 64

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

\[ y(x)\to c_1 \left (\frac {27 x^2}{8}+\frac {1}{x^2}-\frac {9 x}{2}-\frac {3}{x}+\frac {9}{2}\right )+c_2 \left (\frac {27 x^5}{280}-\frac {9 x^4}{40}+\frac {9 x^3}{20}-\frac {3 x^2}{4}+x\right ) \]