Internal problem ID [6577]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. Section 6.2
SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number: 25 expansion at 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\cos \left (x \right ) y^{\prime \prime }+y^{\prime }+5 y=0} \] With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 860
Order:=8; dsolve(cos(x)*diff(y(x),x$2)+diff(y(x),x)+5*y(x)=0,y(x),type='series',x=1);
\[ y \left (x \right ) = \left (1-\frac {5 \sec \left (1\right ) \left (x -1\right )^{2}}{2}-\frac {5 \left (\sin \left (1\right )-1\right ) \sec \left (1\right )^{2} \left (x -1\right )^{3}}{6}+\frac {5 \sec \left (1\right )^{3} \left (\cos \left (1\right )^{2}+5 \cos \left (1\right )+3 \sin \left (1\right )-3\right ) \left (x -1\right )^{4}}{24}+\frac {\sec \left (1\right )^{4} \left (12+\left (\cos \left (1\right )^{2}+20 \cos \left (1\right )-12\right ) \sin \left (1\right )-7 \cos \left (1\right )^{2}-10 \cos \left (1\right )\right ) \left (x -1\right )^{5}}{24}-\frac {\sec \left (1\right )^{5} \left (\cos \left (1\right )^{4}+55 \cos \left (1\right )^{3}+\left (15 \sin \left (1\right )-20\right ) \cos \left (1\right )^{2}+\left (75 \sin \left (1\right )-105\right ) \cos \left (1\right )-60 \sin \left (1\right )+60\right ) \left (x -1\right )^{6}}{144}+\frac {\sec \left (1\right )^{6} \left (360-\left (\cos \left (1\right )^{4}+130 \cos \left (1\right )^{3}+75 \cos \left (1\right )^{2}-660 \cos \left (1\right )+360\right ) \sin \left (1\right )+31 \cos \left (1\right )^{4}+365 \cos \left (1\right )^{3}-255 \cos \left (1\right )^{2}-540 \cos \left (1\right )\right ) \left (x -1\right )^{7}}{1008}\right ) y \left (1\right )+\left (x -1-\frac {\sec \left (1\right ) \left (x -1\right )^{2}}{2}-\frac {5 \left (\cos \left (1\right )+\frac {\sin \left (1\right )}{5}-\frac {1}{5}\right ) \sec \left (1\right )^{2} \left (x -1\right )^{3}}{6}+\frac {5 \left (\frac {\cos \left (1\right )^{2}}{5}+\left (-2 \sin \left (1\right )+2\right ) \cos \left (1\right )+\frac {3 \sin \left (1\right )}{5}-\frac {3}{5}\right ) \sec \left (1\right )^{3} \left (x -1\right )^{4}}{24}+\frac {\sec \left (1\right )^{4} \left (\left (\cos \left (1\right )^{2}+45 \cos \left (1\right )-12\right ) \sin \left (1\right )+15 \cos \left (1\right )^{3}+18 \cos \left (1\right )^{2}-45 \cos \left (1\right )+12\right ) \left (x -1\right )^{5}}{120}-\frac {\sec \left (1\right )^{5} \left (\frac {\cos \left (1\right )^{4}}{5}+\left (-4 \sin \left (1\right )+28\right ) \cos \left (1\right )^{3}+\left (-27 \sin \left (1\right )+6\right ) \cos \left (1\right )^{2}+\left (48 \sin \left (1\right )-48\right ) \cos \left (1\right )-12 \sin \left (1\right )+12\right ) \left (x -1\right )^{6}}{144}-\frac {\sec \left (1\right )^{6} \left (\left (\cos \left (1\right )^{4}+375 \cos \left (1\right )^{3}+600 \cos \left (1\right )^{2}-1500 \cos \left (1\right )+360\right ) \sin \left (1\right )+25 \cos \left (1\right )^{5}+544 \cos \left (1\right )^{4}-1000 \cos \left (1\right )^{3}-720 \cos \left (1\right )^{2}+1500 \cos \left (1\right )-360\right ) \left (x -1\right )^{7}}{5040}\right ) D\left (y \right )\left (1\right )+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 1808
AsymptoticDSolveValue[Cos[x]*y''[x]+y'[x]+5*y[x]==0,y[x],{x,1,7}]
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