12.14.4 problem 1

Internal problem ID [1945]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 1
Date solved : Monday, January 27, 2025 at 05:38:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }+x \left (5 x^{2}+3 x +3\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 317

Order:=6; 
dsolve(2*x^2*(1+x+x^2)*diff(y(x),x$2)+x*(3+3*x+5*x^2)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{\frac {i \sqrt {7}}{4}} \left (1+\frac {1}{2+i \sqrt {7}} x +\frac {1}{4} \frac {11-i \sqrt {7}}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right )} x^{2}-\frac {1}{12} \frac {49 i \sqrt {7}+89}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right )} x^{3}+\frac {1}{48} \frac {395 i \sqrt {7}-1553}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right )} x^{4}+\frac {1}{240} \frac {42423 i \sqrt {7}+45275}{\left (2+i \sqrt {7}\right ) \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+6\right ) \left (i \sqrt {7}+8\right ) \left (i \sqrt {7}+10\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \,x^{-\frac {i \sqrt {7}}{4}} \left (1+\frac {1}{2-i \sqrt {7}} x +\frac {-11-i \sqrt {7}}{-4+24 i \sqrt {7}} x^{2}+\frac {49 \sqrt {7}+89 i}{432 i-444 \sqrt {7}} x^{3}-\frac {1}{48} \frac {395 i \sqrt {7}+1553}{\left (\sqrt {7}+2 i\right ) \left (\sqrt {7}+4 i\right ) \left (\sqrt {7}+6 i\right ) \left (\sqrt {7}+8 i\right )} x^{4}+\frac {-42423 \sqrt {7}-45275 i}{1749600 i-492720 \sqrt {7}} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{1}/{4}}} \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 4838

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

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