56.4.39 problem 36

Internal problem ID [8928]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 36
Date solved : Monday, January 27, 2025 at 05:20:09 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 60

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

\[ y = \ln \left (x \right ) \left (x +2 x^{2}+3 x^{3}+4 x^{4}+5 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+3 x +5 x^{2}+7 x^{3}+9 x^{4}+11 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (x^4+x^3+x^2+\left (4 x^3+3 x^2+2 x+1\right ) x \log (x)+x+1\right )+c_2 \left (5 x^5+4 x^4+3 x^3+2 x^2+x\right ) \]