34.5.2 problem 6

Internal problem ID [6070]
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 : 6
Date solved : Monday, January 27, 2025 at 01:34:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 42

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

\[ y = c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\ln \left (x \right ) \left (\frac {1}{2} x +\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1-\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{32} x^{3}+\frac {5}{384} x^{4}+\frac {7}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 43

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

\[ y(x)\to c_1 \left (\frac {1}{384} \left (5 x^4+12 x^3+48 x^2-768 x+384\right )+\frac {1}{2} x \log (x)\right )+c_2 x \]