34.3.13 problem 15

Internal problem ID [6052]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XIV. page 177
Problem number : 15
Date solved : Monday, January 27, 2025 at 01:34:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) 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: 41

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+(x+x^2)*diff(y(x),x)+(x-9)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{3} \left (1-\frac {4}{7} x +\frac {5}{28} x^{2}-\frac {5}{126} x^{3}+\frac {1}{144} x^{4}-\frac {1}{990} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-86400+34560 x -4320 x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.029 (sec). Leaf size: 60

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+(x+x^2)*D[y[x],x]+(x-9)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}-\frac {2}{5 x^2}+\frac {1}{20 x}\right )+c_2 \left (\frac {x^7}{144}-\frac {5 x^6}{126}+\frac {5 x^5}{28}-\frac {4 x^4}{7}+x^3\right ) \]