Internal problem ID [2422]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 43, page 209
Problem number: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x +12\right ) y=x^{2}+x} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 379
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x+12)*y(x)=x^2+x,y(x),type='series',x=0);
\[ y \left (x \right ) = c_{2} x^{2 i \sqrt {3}} \left (1+\frac {1}{-4 i \sqrt {3}-1} x -\frac {1}{4} \frac {1}{\left (i-2 \sqrt {3}\right ) \left (-4 \sqrt {3}+i\right )} x^{2}+\frac {1}{48} \frac {1}{\left (i-2 \sqrt {3}\right ) \left (-4 \sqrt {3}+i\right ) \left (i \sqrt {3}+\frac {3}{4}\right )} x^{3}+\frac {1}{768} \frac {1}{\left (-\sqrt {3}+\frac {3 i}{4}\right ) \left (-i+2 \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \left (-4 \sqrt {3}+i\right )} x^{4}+\frac {1}{15360} \frac {1}{\left (\sqrt {3}-\frac {3 i}{4}\right ) \left (-i+2 \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \left (-4 \sqrt {3}+i\right ) \left (i \sqrt {3}+\frac {5}{4}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-2 i \sqrt {3}} \left (1+\frac {1}{4 i \sqrt {3}-1} x -\frac {1}{4} \frac {1}{\left (2 \sqrt {3}+i\right ) \left (4 \sqrt {3}+i\right )} x^{2}-\frac {1}{48} \frac {1}{\left (2 \sqrt {3}+i\right ) \left (4 \sqrt {3}+i\right ) \left (i \sqrt {3}-\frac {3}{4}\right )} x^{3}+\frac {1}{192} \frac {1}{\left (2 \sqrt {3}+i\right ) \left (3 i+4 \sqrt {3}\right ) \left (\sqrt {3}+i\right ) \left (4 \sqrt {3}+i\right )} x^{4}+\frac {1}{960} \frac {1}{\left (4 i \sqrt {3}-3\right ) \left (2 \sqrt {3}+i\right ) \left (\sqrt {3}+i\right ) \left (5 i+4 \sqrt {3}\right ) \left (4 \sqrt {3}+i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (\frac {1}{13}+\frac {3}{52} x -\frac {1}{364} x^{2}+\frac {1}{10192} x^{3}-\frac {1}{377104} x^{4}+\operatorname {O}\left (x^{5}\right )\right ) \]
✓ Solution by Mathematica
Time used: 1.224 (sec). Leaf size: 704
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x+12)*y[x]==x^2+x,y[x],{x,0,5}]
\[ y(x)\to -\frac {\left (518 \left (139 \sqrt {3}-100 i\right ) x^5-2555 \left (929 \sqrt {3}+1053 i\right ) x^4-46720 \left (121 \sqrt {3}-2726 i\right ) x^3+9320640 \left (125 \sqrt {3}-72 i\right ) x^2-55923840 \left (97 \sqrt {3}+257 i\right ) x-1826845440 \left (5 \sqrt {3}+6 i\right )\right ) \left (\frac {i x^5}{720960 \sqrt {3}-2865600 i}-\frac {i x^4}{41280 \sqrt {3}-15552 i}+\frac {i x^3}{888 \sqrt {3}+1692 i}+\frac {i x^2}{24 \sqrt {3}-92 i}-\frac {x}{1-4 i \sqrt {3}}+1\right ) x}{3131735040 \left (84 \sqrt {3}+49 i\right )}+\frac {\left (141+74 i \sqrt {3}\right ) \left (4 \sqrt {3}-i\right ) \left (6 \sqrt {3}+23 i\right ) \left (215 \sqrt {3}+81 i\right ) \left (751 \sqrt {3}+2985 i\right ) \left (-\frac {i x^5}{720960 \sqrt {3}+2865600 i}+\frac {i x^4}{41280 \sqrt {3}+15552 i}-\frac {i x^3}{888 \sqrt {3}-1692 i}-\frac {i x^2}{24 \sqrt {3}+92 i}-\frac {x}{1+4 i \sqrt {3}}+1\right ) \left (259 \left (3633339 \sqrt {3}-14076539 i\right ) x^5+35 \left (1258384403 \sqrt {3}+3533247843 i\right ) x^4-2240 \left (972417857 \sqrt {3}-84278905 i\right ) x^3+480 \left (25781300141 \sqrt {3}-123179074077 i\right ) x^2+40320 \left (5977658307 \sqrt {3}+7170676999 i\right ) x+13440 \left (13382594611 \sqrt {3}+35444283093 i\right )\right ) x}{2233732632339984513454080 \sqrt {3} \left (-49-84 i \sqrt {3}\right )}+c_2 \left (-\frac {i x^5}{720960 \sqrt {3}+2865600 i}+\frac {i x^4}{41280 \sqrt {3}+15552 i}-\frac {i x^3}{888 \sqrt {3}-1692 i}-\frac {i x^2}{24 \sqrt {3}+92 i}-\frac {x}{1+4 i \sqrt {3}}+1\right ) x^{2 i \sqrt {3}}+c_1 \left (\frac {i x^5}{720960 \sqrt {3}-2865600 i}-\frac {i x^4}{41280 \sqrt {3}-15552 i}+\frac {i x^3}{888 \sqrt {3}+1692 i}+\frac {i x^2}{24 \sqrt {3}-92 i}-\frac {x}{1-4 i \sqrt {3}}+1\right ) x^{-2 i \sqrt {3}} \]