Internal problem ID [5824]
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]]
Solve \begin {gather*} \boxed {\cos \relax (x ) y^{\prime \prime }+y^{\prime }+5 y=0} \end {gather*} With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.031 (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 \relax (x ) = \left (1-\frac {5 \left (x -1\right )^{2}}{2 \cos \relax (1)}+\frac {\left (-5 \sin \relax (1)+5\right ) \left (x -1\right )^{3}}{6 \cos \relax (1)^{2}}+\frac {\left (5 \left (\cos ^{2}\relax (1)\right )+15 \sin \relax (1)+25 \cos \relax (1)-15\right ) \left (x -1\right )^{4}}{24 \cos \relax (1)^{3}}+\frac {\left (\left (5 \sin \relax (1)-35\right ) \left (\cos ^{2}\relax (1)\right )+\left (100 \sin \relax (1)-50\right ) \cos \relax (1)-60 \sin \relax (1)+60\right ) \left (x -1\right )^{5}}{120 \cos \relax (1)^{4}}+\frac {\left (-5 \left (\cos ^{4}\relax (1)\right )-275 \left (\cos ^{3}\relax (1)\right )+\left (-75 \sin \relax (1)+100\right ) \left (\cos ^{2}\relax (1)\right )+\left (-375 \sin \relax (1)+525\right ) \cos \relax (1)+300 \sin \relax (1)-300\right ) \left (x -1\right )^{6}}{720 \cos \relax (1)^{5}}+\frac {\left (\left (-5 \sin \relax (1)+155\right ) \left (\cos ^{4}\relax (1)\right )+\left (-650 \sin \relax (1)+1825\right ) \left (\cos ^{3}\relax (1)\right )+\left (-375 \sin \relax (1)-1275\right ) \left (\cos ^{2}\relax (1)\right )+\left (3300 \sin \relax (1)-2700\right ) \cos \relax (1)-1800 \sin \relax (1)+1800\right ) \left (x -1\right )^{7}}{5040 \cos \relax (1)^{6}}\right ) y \relax (1)+\left (x -1-\frac {\left (x -1\right )^{2}}{2 \cos \relax (1)}+\frac {\left (-\sin \relax (1)-5 \cos \relax (1)+1\right ) \left (x -1\right )^{3}}{6 \cos \relax (1)^{2}}+\frac {\left (\cos ^{2}\relax (1)+\left (-10 \sin \relax (1)+10\right ) \cos \relax (1)+3 \sin \relax (1)-3\right ) \left (x -1\right )^{4}}{24 \cos \relax (1)^{3}}+\frac {\left (15 \left (\cos ^{3}\relax (1)\right )+\left (\sin \relax (1)+18\right ) \left (\cos ^{2}\relax (1)\right )+\left (45 \sin \relax (1)-45\right ) \cos \relax (1)-12 \sin \relax (1)+12\right ) \left (x -1\right )^{5}}{120 \cos \relax (1)^{4}}+\frac {\left (-\left (\cos ^{4}\relax (1)\right )+\left (20 \sin \relax (1)-140\right ) \left (\cos ^{3}\relax (1)\right )+\left (135 \sin \relax (1)-30\right ) \left (\cos ^{2}\relax (1)\right )+\left (-240 \sin \relax (1)+240\right ) \cos \relax (1)+60 \sin \relax (1)-60\right ) \left (x -1\right )^{6}}{720 \cos \relax (1)^{5}}+\frac {\left (-25 \left (\cos ^{5}\relax (1)\right )+\left (-\sin \relax (1)-544\right ) \left (\cos ^{4}\relax (1)\right )+\left (-375 \sin \relax (1)+1000\right ) \left (\cos ^{3}\relax (1)\right )+\left (-600 \sin \relax (1)+720\right ) \left (\cos ^{2}\relax (1)\right )+\left (1500 \sin \relax (1)-1500\right ) \cos \relax (1)-360 \sin \relax (1)+360\right ) \left (x -1\right )^{7}}{5040 \cos \relax (1)^{6}}\right ) D\relax (y )\relax (1)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 1808
AsymptoticDSolveValue[Cos[x]*y''[x]+y'[x]+5*y[x]==0,y[x],{x,1,7}]
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