15.23.7 problem 7

Internal problem ID [3357]
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 : 7
Date solved : Monday, January 27, 2025 at 07:34:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 45

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

\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {5}{4} x +\frac {5}{96} x^{2}-\frac {11}{1152} x^{3}+\frac {341}{129024} x^{4}-\frac {20119}{23224320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{3} x -\frac {1}{30} x^{2}+\frac {1}{126} x^{3}-\frac {11}{4536} x^{4}+\frac {19}{22680} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 86

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

\[ y(x)\to c_1 x \left (\frac {19 x^5}{22680}-\frac {11 x^4}{4536}+\frac {x^3}{126}-\frac {x^2}{30}+\frac {x}{3}+1\right )+c_2 \sqrt {x} \left (-\frac {20119 x^5}{23224320}+\frac {341 x^4}{129024}-\frac {11 x^3}{1152}+\frac {5 x^2}{96}+\frac {5 x}{4}+1\right ) \]