15.23.4 problem 4

Internal problem ID [3354]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 4
Date solved : Monday, January 27, 2025 at 07:34:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 45

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

\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x +\frac {1}{35} x^{2}+\frac {1}{315} x^{3}+\frac {1}{3465} x^{4}+\frac {1}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 86

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

\[ y(x)\to c_1 x \left (\frac {x^5}{45045}+\frac {x^4}{3465}+\frac {x^3}{315}+\frac {x^2}{35}+\frac {x}{5}+1\right )+\frac {c_2 \left (\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right )}{\sqrt {x}} \]