54.4.1 problem 1

Internal problem ID [8585]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots Nonintegral. Exercises page 365
Problem number : 1
Date solved : Monday, January 27, 2025 at 04:16:49 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 2 x \left (1+x \right ) y^{\prime \prime }+3 \left (1+x \right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 40

Order:=8; 
dsolve(2*x*(x+1)*diff(y(x),x$2)+3*(x+1)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+x +\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}+c_{2} \left (1+\frac {1}{3} x -\frac {1}{15} x^{2}+\frac {1}{35} x^{3}-\frac {1}{63} x^{4}+\frac {1}{99} x^{5}-\frac {1}{143} x^{6}+\frac {1}{195} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 67

AsymptoticDSolveValue[2*x*(x+1)*D[y[x],{x,2}]+3*(x+1)*D[y[x],x]-y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^7}{195}-\frac {x^6}{143}+\frac {x^5}{99}-\frac {x^4}{63}+\frac {x^3}{35}-\frac {x^2}{15}+\frac {x}{3}+1\right )+\frac {c_2 (x+1)}{\sqrt {x}} \]