12.11.11 problem 22

Internal problem ID [1850]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 22
Date solved : Monday, January 27, 2025 at 05:37:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (2 x +1\right ) y^{\prime }-\left (4+6 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 47

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

\[ y = c_1 \,x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (\ln \left (x \right ) \left (576 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+192 x -288 x^{2}+576 x^{3}-576 x^{4}-576 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 48

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

\[ y(x)\to c_2 x^2+c_1 \left (\frac {3 x^4-12 x^3+6 x^2-4 x+3}{3 x^2}-4 x^2 \log (x)\right ) \]