34.5.4 problem 9

Internal problem ID [6072]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XVI. page 220
Problem number : 9
Date solved : Monday, January 27, 2025 at 01:34:53 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\frac {\left (1-2 x \right ) y^{\prime }}{3}+\frac {20 y}{9}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 36

Order:=6; 
dsolve(x*(1-x)*diff(y(x),x$2)+1/3*(1-2*x)*diff(y(x),x)+20/9*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{2}/{3}} \left (1-\frac {6}{5} x +\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {20}{3} x +\frac {35}{9} x^{2}+\frac {50}{81} x^{3}+\frac {65}{243} x^{4}+\frac {112}{729} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 57

AsymptoticDSolveValue[x*(1-x)*D[y[x],{x,2}]+1/3*(1-2*x)*D[y[x],x]+20/9*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (1-\frac {6 x}{5}\right ) x^{2/3}+c_2 \left (\frac {112 x^5}{729}+\frac {65 x^4}{243}+\frac {50 x^3}{81}+\frac {35 x^2}{9}-\frac {20 x}{3}+1\right ) \]