34.5.1 problem 5

Internal problem ID [6069]
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 : 5
Date solved : Monday, January 27, 2025 at 01:34:49 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 34

Order:=6; 
dsolve(x*(1-x)*diff(y(x),x$2)+(3/2-2*x)*diff(y(x),x)-1/4*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 \left (1+\frac {1}{6} x +\frac {3}{40} x^{2}+\frac {5}{112} x^{3}+\frac {35}{1152} x^{4}+\frac {63}{2816} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 50

AsymptoticDSolveValue[x*(1-x)*D[y[x],{x,2}]+(3/2-2*x)*D[y[x],x]-1/4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {63 x^5}{2816}+\frac {35 x^4}{1152}+\frac {5 x^3}{112}+\frac {3 x^2}{40}+\frac {x}{6}+1\right )+\frac {c_2}{\sqrt {x}} \]