54.5.5 problem 5

Internal problem ID [8620]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 5
Date solved : Monday, January 27, 2025 at 04:17:37 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (1+x \right ) y^{\prime \prime }+\left (1+5 x \right ) y^{\prime }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 52

Order:=8; 
dsolve(x*(1+x)*diff(y(x),x$2)+(1+5*x)*diff(y(x),x)+3*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-3 x +6 x^{2}-10 x^{3}+15 x^{4}-21 x^{5}+28 x^{6}-36 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {11}{2} x^{2}+\frac {21}{2} x^{3}-17 x^{4}+25 x^{5}-\frac {69}{2} x^{6}+\frac {91}{2} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 125

AsymptoticDSolveValue[x*(1+x)*D[y[x],{x,2}]+(1+5*x)*D[y[x],x]+3*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (-36 x^7+28 x^6-21 x^5+15 x^4-10 x^3+6 x^2-3 x+1\right )+c_2 \left (\frac {91 x^7}{2}-\frac {69 x^6}{2}+25 x^5-17 x^4+\frac {21 x^3}{2}-\frac {11 x^2}{2}+\left (-36 x^7+28 x^6-21 x^5+15 x^4-10 x^3+6 x^2-3 x+1\right ) \log (x)+2 x\right ) \]