68.1.9 problem Problem 1.7

Internal problem ID [14138]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS. Problems page 28
Problem number : Problem 1.7
Date solved : Tuesday, January 28, 2025 at 06:15:46 AM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (1-5 x \right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 44

Order:=6; 
dsolve(x*(1-x)*diff(y(x),x$2)+(1-5*x)*diff(y(x),x)-4*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1+4 x +9 x^{2}+16 x^{3}+25 x^{4}+36 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-4\right ) x -12 x^{2}-24 x^{3}-40 x^{4}-60 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 87

AsymptoticDSolveValue[x*(1-x)*D[y[x],{x,2}]+(1-5*x)*D[y[x],x]-4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (36 x^5+25 x^4+16 x^3+9 x^2+4 x+1\right )+c_2 \left (-60 x^5-40 x^4-24 x^3-12 x^2+\left (36 x^5+25 x^4+16 x^3+9 x^2+4 x+1\right ) \log (x)-4 x\right ) \]