41.1.8 problem Ex. 8(i), page 258

Internal problem ID [6821]
Book : A treatise on Differential Equations by A. R. Forsyth. 6th edition. 1929. Macmillan Co. ltd. New York, reprinted 1956
Section : Chapter VI. Note I. Integration of linear equations in series by the method of Frobenius. page 243
Problem number : Ex. 8(i), page 258
Date solved : Monday, January 27, 2025 at 02:31:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 407

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

\[ y = \sqrt {x}\, \left (c_1 \,x^{-\frac {\sqrt {1-16 a}}{2}} \left (1+4 \frac {1}{-1+\sqrt {1-16 a}} x +8 \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right )} x^{2}+\frac {32}{3} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right )} x^{3}+\frac {32}{3} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right )} x^{4}+\frac {128}{15} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right ) \left (-5+\sqrt {1-16 a}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\frac {\sqrt {1-16 a}}{2}} \left (1-4 \frac {1}{1+\sqrt {1-16 a}} x +8 \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right )} x^{2}-\frac {32}{3} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right )} x^{3}+\frac {32}{3} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right )} x^{4}-\frac {128}{15} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right ) \left (5+\sqrt {1-16 a}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 1356

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

\[ y(x)\to \left (-\frac {1024 x^5}{\left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+2\right )+4 a\right ) \left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+2\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+3\right )+4 a\right ) \left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+3\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+4\right )+4 a\right ) \left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+4\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+5\right )+4 a\right ) \left (\frac {1}{2} \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (1-\sqrt {1-16 a}\right )+4 a\right )}+\frac {256 x^4}{\left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+2\right )+4 a\right ) \left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+2\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+3\right )+4 a\right ) \left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+3\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+4\right )+4 a\right ) \left (\frac {1}{2} \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (1-\sqrt {1-16 a}\right )+4 a\right )}-\frac {64 x^3}{\left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+2\right )+4 a\right ) \left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+2\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+3\right )+4 a\right ) \left (\frac {1}{2} \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (1-\sqrt {1-16 a}\right )+4 a\right )}+\frac {16 x^2}{\left (\left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+2\right )+4 a\right ) \left (\frac {1}{2} \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (1-\sqrt {1-16 a}\right )+4 a\right )}-\frac {4 x}{\frac {1}{2} \left (\frac {1}{2} \left (1-\sqrt {1-16 a}\right )+1\right ) \left (1-\sqrt {1-16 a}\right )+4 a}+1\right ) c_2 x^{\frac {1}{2} \left (1-\sqrt {1-16 a}\right )}+\left (-\frac {1024 x^5}{\left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+2\right )+4 a\right ) \left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+2\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+3\right )+4 a\right ) \left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+3\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+4\right )+4 a\right ) \left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+4\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+5\right )+4 a\right ) \left (\frac {1}{2} \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\sqrt {1-16 a}+1\right )+4 a\right )}+\frac {256 x^4}{\left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+2\right )+4 a\right ) \left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+2\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+3\right )+4 a\right ) \left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+3\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+4\right )+4 a\right ) \left (\frac {1}{2} \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\sqrt {1-16 a}+1\right )+4 a\right )}-\frac {64 x^3}{\left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+2\right )+4 a\right ) \left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+2\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+3\right )+4 a\right ) \left (\frac {1}{2} \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\sqrt {1-16 a}+1\right )+4 a\right )}+\frac {16 x^2}{\left (\left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+2\right )+4 a\right ) \left (\frac {1}{2} \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\sqrt {1-16 a}+1\right )+4 a\right )}-\frac {4 x}{\frac {1}{2} \left (\frac {1}{2} \left (\sqrt {1-16 a}+1\right )+1\right ) \left (\sqrt {1-16 a}+1\right )+4 a}+1\right ) c_1 x^{\frac {1}{2} \left (\sqrt {1-16 a}+1\right )} \]