15.23.8 problem 8

Internal problem ID [3358]
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 : 8
Date solved : Monday, January 27, 2025 at 07:34:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }-3 \left (x^{2}+x \right ) y^{\prime }+\left (3 x +2\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: 47

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

\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {3}{2} x +\frac {9}{8} x^{2}+\frac {9}{16} x^{3}+\frac {27}{128} x^{4}+\frac {81}{1280} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+\frac {3}{5} x +\frac {9}{35} x^{2}+\frac {3}{35} x^{3}+\frac {9}{385} x^{4}+\frac {27}{5005} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 88

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

\[ y(x)\to c_1 \left (\frac {27 x^5}{5005}+\frac {9 x^4}{385}+\frac {3 x^3}{35}+\frac {9 x^2}{35}+\frac {3 x}{5}+1\right ) x^2+c_2 \left (\frac {81 x^5}{1280}+\frac {27 x^4}{128}+\frac {9 x^3}{16}+\frac {9 x^2}{8}+\frac {3 x}{2}+1\right ) \sqrt {x} \]