Internal problem ID [5836]
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. 6.3 SOLUTIONS ABOUT
SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x -5\right )^{2} y^{\prime \prime }+4 x y^{\prime }+\left (x^{2}-25\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.036 (sec). Leaf size: 2223
Order:=8; dsolve(x^2*(x-5)^2*diff(y(x),x$2)+4*x*diff(y(x),x)+(x^2-25)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {21}{50}-\frac {\sqrt {2941}}{50}} \left (1+\frac {-1166-4 \sqrt {2941}}{-3125+125 \sqrt {2941}} x -\frac {9}{15625} \frac {879 \sqrt {2941}-79709}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right )} x^{2}+\frac {\frac {15291084 \sqrt {2941}}{1953125}-\frac {906742764}{1953125}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right )} x^{3}-\frac {12}{244140625} \frac {2200649681 \sqrt {2941}-122814219551}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right )} x^{4}+\frac {\frac {181292058002304 \sqrt {2941}}{152587890625}-\frac {10008934775328384}{152587890625}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right )} x^{5}+\frac {\frac {250187169310576512 \sqrt {2941}}{19073486328125}-\frac {13371141904684696752}{19073486328125}}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right ) \left (-150+\sqrt {2941}\right )} x^{6}-\frac {96}{16689300537109375} \frac {381820145596656632404 \sqrt {2941}-20689947387639015669859}{\left (-25+\sqrt {2941}\right ) \left (-50+\sqrt {2941}\right ) \left (-75+\sqrt {2941}\right ) \left (-100+\sqrt {2941}\right ) \left (-125+\sqrt {2941}\right ) \left (-150+\sqrt {2941}\right ) \left (-175+\sqrt {2941}\right )} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {21}{50}+\frac {\sqrt {2941}}{50}} \left (1+\frac {1166-4 \sqrt {2941}}{125 \sqrt {2941}+3125} x +\frac {\frac {7911 \sqrt {2941}}{15625}+\frac {717381}{15625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right )} x^{2}+\frac {\frac {15291084 \sqrt {2941}}{1953125}+\frac {906742764}{1953125}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right )} x^{3}+\frac {\frac {26407796172 \sqrt {2941}}{244140625}+\frac {1473770634612}{244140625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right )} x^{4}+\frac {\frac {181292058002304 \sqrt {2941}}{152587890625}+\frac {10008934775328384}{152587890625}}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right )} x^{5}-\frac {48}{19073486328125} \frac {5212232693970344 \sqrt {2941}+278565456347597849}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right ) \left (150+\sqrt {2941}\right )} x^{6}-\frac {96}{16689300537109375} \frac {381820145596656632404 \sqrt {2941}+20689947387639015669859}{\left (\sqrt {2941}+25\right ) \left (50+\sqrt {2941}\right ) \left (\sqrt {2941}+75\right ) \left (100+\sqrt {2941}\right ) \left (125+\sqrt {2941}\right ) \left (150+\sqrt {2941}\right ) \left (175+\sqrt {2941}\right )} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 22488
AsymptoticDSolveValue[x^2*(x-5)^2*y''[x]+4*x*y'[x]+(x^2-25)*y[x]==0,y[x],{x,0,7}]
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