15.23.5 problem 5

Internal problem ID [3355]
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 : 5
Date solved : Monday, January 27, 2025 at 07:34:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+5 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: 0.014 (sec). Leaf size: 47

Order:=6; 
dsolve(2*x^2*diff(y(x),x$2)+5*x*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {c_{1} \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 86

AsymptoticDSolveValue[2*x^2*D[y[x],{x,2}]+5*x*D[y[x],x]+(1+x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {x^5}{1247400}+\frac {x^4}{22680}-\frac {x^3}{630}+\frac {x^2}{30}-\frac {x}{3}+1\right )}{\sqrt {x}}+\frac {c_2 \left (-\frac {x^5}{113400}+\frac {x^4}{2520}-\frac {x^3}{90}+\frac {x^2}{6}-x+1\right )}{x} \]