7.16.13 problem 13

Internal problem ID [510]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 13
Date solved : Monday, January 27, 2025 at 02:54:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 60

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

\[ y = x \left (c_1 \,x^{2} \left (1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (8 x^{2}-16 x^{3}+16 x^{4}-\frac {32}{3} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-4 x +12 x^{2}-8 x^{3}+\frac {32}{9} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 73

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

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