2.17.120 Problems 11901 to 12000

Problem 11901

ODE	\[ \boxed {\left (x^{2}-3 x \right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{3}} \left (1+\frac {17 x}{36}+\frac {1241 x^{2}}{7128}+\frac {80665 x^{3}}{1347192}+\frac {972725 x^{4}}{48498912}+\frac {5797441 x^{5}}{872980416}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {x}{2}-\frac {x^{2}}{2}-\frac {5 x^{3}}{24}-\frac {25 x^{4}}{336}-\frac {17 x^{5}}{672}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {5}{3}} \left (1+\frac {17}{36} x +\frac {1241}{7128} x^{2}+\frac {80665}{1347192} x^{3}+\frac {972725}{48498912} x^{4}+\frac {5797441}{872980416} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{2} x -\frac {1}{2} x^{2}-\frac {5}{24} x^{3}-\frac {25}{336} x^{4}-\frac {17}{672} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 11902

ODE	\[ \boxed {\left (x^{3}+x^{2}\right ) y^{\prime \prime }+\left (x^{2}-2 x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}+\frac {i \sqrt {7}}{2}} \left (1-\frac {\left (3+i \sqrt {7}\right )^{2} x}{4+4 i \sqrt {7}}+\frac {\left (-4 \sqrt {7}-12 i\right ) x^{2}}{\left (-\sqrt {7}+i\right ) \left (i \sqrt {7}+2\right )}-\frac {7 \left (3+i \sqrt {7}\right )^{2} \left (i \sqrt {7}+5\right )^{2} x^{3}}{192 \left (i \sqrt {7}+1\right ) \left (i \sqrt {7}+2\right )}+\frac {3 \left (3+i \sqrt {7}\right ) \left (i \sqrt {7}+5\right )^{2} \left (i \sqrt {7}+7\right )^{2} \left (i \sqrt {7}+\frac {37}{9}\right ) x^{4}}{1024 \left (i \sqrt {7}+4\right ) \left (i \sqrt {7}+1\right ) \left (i \sqrt {7}+2\right )}+\frac {11 \left (-3 i+\sqrt {7}\right ) \left (i \sqrt {7}+5\right ) \left (i \sqrt {7}+7\right )^{2} \left (i \sqrt {7}+\frac {57}{11}\right ) \left (i \sqrt {7}+9\right )^{2} x^{5}}{61440 \left (i \sqrt {7}+4\right ) \left (-\sqrt {7}+i\right ) \left (i \sqrt {7}+2\right )}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {3}{2}-\frac {i \sqrt {7}}{2}} \left (1-\frac {\left (3-i \sqrt {7}\right )^{2} x}{4-4 i \sqrt {7}}+\frac {\left (-4 \sqrt {7}+12 i\right ) x^{2}}{\left (-\sqrt {7}-i\right ) \left (-i \sqrt {7}+2\right )}-\frac {7 \left (3-i \sqrt {7}\right )^{2} \left (-i \sqrt {7}+5\right )^{2} x^{3}}{192 \left (-i \sqrt {7}+1\right ) \left (-i \sqrt {7}+2\right )}+\frac {3 \left (3-i \sqrt {7}\right ) \left (-i \sqrt {7}+5\right )^{2} \left (-i \sqrt {7}+7\right )^{2} \left (-i \sqrt {7}+\frac {37}{9}\right ) x^{4}}{1024 \left (-i \sqrt {7}+4\right ) \left (-i \sqrt {7}+1\right ) \left (-i \sqrt {7}+2\right )}+\frac {11 \left (3 i+\sqrt {7}\right ) \left (-i \sqrt {7}+5\right ) \left (-i \sqrt {7}+7\right )^{2} \left (-i \sqrt {7}+\frac {57}{11}\right ) \left (-i \sqrt {7}+9\right )^{2} x^{5}}{61440 \left (-i \sqrt {7}+4\right ) \left (-\sqrt {7}-i\right ) \left (-i \sqrt {7}+2\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {3}{2}} \left (c_{2} x^{\frac {i \sqrt {7}}{2}} \left (1+\frac {3 \sqrt {7}-i}{-2 \sqrt {7}+2 i} x +\frac {-4 \sqrt {7}-12 i}{\left (-\sqrt {7}+i\right ) \left (i \sqrt {7}+2\right )} x^{2}+\frac {224}{3} \frac {1}{\left (\sqrt {7}-2 i\right ) \left (-\sqrt {7}+i\right ) \left (3+i \sqrt {7}\right )} x^{3}+\frac {84 \sqrt {7}-\frac {1036 i}{3}}{\left (-\sqrt {7}+i\right ) \left (i \sqrt {7}+2\right ) \left (3+i \sqrt {7}\right ) \left (4+i \sqrt {7}\right )} x^{4}+\frac {\frac {2576 i \sqrt {7}}{3}+\frac {6608}{5}}{\left (-4 i+\sqrt {7}\right ) \left (-\sqrt {7}+i\right ) \left (i \sqrt {7}+2\right ) \left (3+i \sqrt {7}\right ) \left (i \sqrt {7}+5\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {i \sqrt {7}}{2}} \left (1+\frac {-3 \sqrt {7}-i}{2 \sqrt {7}+2 i} x +\frac {12+4 i \sqrt {7}}{5+3 i \sqrt {7}} x^{2}+\frac {224}{3} \frac {1}{\left (i \sqrt {7}-2\right ) \left (\sqrt {7}+3 i\right ) \left (\sqrt {7}+i\right )} x^{3}+\frac {63 i \sqrt {7}-259}{15 i \sqrt {7}-129} x^{4}+\frac {-1239 i-805 \sqrt {7}}{675 i+255 \sqrt {7}} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 11903

ODE	\[ \boxed {\left (x^{4}-2 x^{3}+x^{2}\right ) y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }+x^{2} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 11904

ODE	\[ \boxed {\left (x^{5}+x^{4}-6 x^{3}\right ) y^{\prime \prime }+y^{\prime } x^{2}+y \left (x -2\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 11905

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+y^{\prime } x +y \left (x^{2}-1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{14}+\frac {x^{4}}{616}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{40}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1-\frac {1}{14} x^{2}+\frac {1}{616} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 11906

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+y^{\prime } x +\left (2 x^{2}-3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x^{2}}{9}+\frac {x^{4}}{234}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x^{2}-\frac {x^{4}}{6}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-\frac {1}{9} x^{2}+\frac {1}{234} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+x^{2}-\frac {1}{6} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 11907

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (x^{2}+\frac {8}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {4}{3}} \left (1-\frac {3 x^{2}}{16}+\frac {9 x^{4}}{896}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {2}{3}} \left (1-\frac {3 x^{2}}{8}+\frac {9 x^{4}}{320}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {2}{3}} \left (1-\frac {3}{8} x^{2}+\frac {9}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {4}{3}} \left (1-\frac {3}{16} x^{2}+\frac {9}{896} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 11908

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (2 x^{2}+\frac {5}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{3}} \left (1-\frac {3 x^{2}}{10}+\frac {9 x^{4}}{320}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {3 x^{2}}{2}+\frac {9 x^{4}}{32}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {3}{2} x^{2}+\frac {9}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {5}{3}} \left (1-\frac {3}{10} x^{2}+\frac {9}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 11909

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {3 x^{2}}{16}+\frac {9 x^{4}}{896}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {3 x^{2}}{8}+\frac {9 x^{4}}{320}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{\frac {2}{3}} \left (1-\frac {3}{16} x^{2}+\frac {9}{896} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1-\frac {3}{8} x^{2}+\frac {9}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{1}}{x^{\frac {1}{3}}} \]

Problem 11910

ODE	\[ \boxed {2 x y^{\prime \prime }+y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {2 x}{3}+\frac {2 x^{2}}{15}-\frac {4 x^{3}}{315}+\frac {2 x^{4}}{2835}-\frac {4 x^{5}}{155925}+O\left (x^{6}\right )\right )+c_{2} \left (1-2 x +\frac {2 x^{2}}{3}-\frac {4 x^{3}}{45}+\frac {2 x^{4}}{315}-\frac {4 x^{5}}{14175}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {2}{3} x +\frac {2}{15} x^{2}-\frac {4}{315} x^{3}+\frac {2}{2835} x^{4}-\frac {4}{155925} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x +\frac {2}{3} x^{2}-\frac {4}{45} x^{3}+\frac {2}{315} x^{4}-\frac {4}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 11911

ODE	\[ \boxed {3 x y^{\prime \prime }-\left (x -2\right ) y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {7 x}{12}+\frac {5 x^{2}}{36}+\frac {13 x^{3}}{648}+\frac {x^{4}}{486}+\frac {19 x^{5}}{116640}+O\left (x^{6}\right )\right )+c_{2} \left (1+x +\frac {3 x^{2}}{10}+\frac {x^{3}}{20}+\frac {x^{4}}{176}+\frac {3 x^{5}}{6160}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1+\frac {7}{12} x +\frac {5}{36} x^{2}+\frac {13}{648} x^{3}+\frac {1}{486} x^{4}+\frac {19}{116640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+x +\frac {3}{10} x^{2}+\frac {1}{20} x^{3}+\frac {1}{176} x^{4}+\frac {3}{6160} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 11912

ODE	\[ \boxed {x y^{\prime \prime }+2 y^{\prime }+x y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{120}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{24}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 11913

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{4}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{120}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{24}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x +c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 11914

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{4}+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x^{3}}{15}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x^{3}}{3}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{15} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-2-\frac {2}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 11915

ODE	\[ \boxed {x y^{\prime \prime }-\left (x^{2}+2\right ) y^{\prime }+x y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+\frac {x^{2}}{5}+\frac {x^{4}}{35}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+\frac {1}{5} x^{2}+\frac {1}{35} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (12+6 x^{2}+\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 11916

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{2}-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x}{2}+\frac {3 x^{2}}{20}-\frac {x^{3}}{30}+\frac {x^{4}}{168}-\frac {x^{5}}{1120}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x}{2}+\frac {x^{3}}{12}-\frac {x^{4}}{24}+\frac {x^{5}}{80}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {1}{2} x +\frac {3}{20} x^{2}-\frac {1}{30} x^{3}+\frac {1}{168} x^{4}-\frac {1}{1120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-6 x +x^{3}-\frac {1}{2} x^{4}+\frac {3}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 11917

ODE	\[ \boxed {\left (2 x^{2}-x \right ) y^{\prime \prime }+\left (2 x -2\right ) y^{\prime }+\left (-2 x^{2}+3 x -2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-4 x +\frac {11 x^{2}}{2}-\frac {7 x^{3}}{3}+\frac {7 x^{4}}{8}-\frac {x^{5}}{5}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-2 x +\frac {7}{2} x^{2}-\frac {4}{3} x^{3}+\frac {13}{24} x^{4}-\frac {7}{60} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 11918

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\frac {3 y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} x +c_{2} \right )+O\left (x^{6}\right ) \]

Problem 11919

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y \left (x -1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{3}+\frac {x^{2}}{24}-\frac {x^{3}}{360}+\frac {x^{4}}{8640}-\frac {x^{5}}{302400}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x \left (1-\frac {x}{3}+\frac {x^{2}}{24}-\frac {x^{3}}{360}+\frac {x^{4}}{8640}-\frac {x^{5}}{302400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+\frac {1+x -\frac {2 x^{3}}{9}+\frac {25 x^{4}}{576}-\frac {157 x^{5}}{43200}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{3} x +\frac {1}{24} x^{2}-\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}-\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{3} x^{3}+\frac {1}{24} x^{4}-\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-2 x +\frac {4}{9} x^{3}-\frac {25}{288} x^{4}+\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 11920

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{3}-x \right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {x^{2}}{4}+\frac {5 x^{4}}{128}+O\left (x^{6}\right )\right )+c_{2} \left (\frac {x^{3} \left (1-\frac {x^{2}}{4}+\frac {5 x^{4}}{128}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{16}+\frac {1-\frac {x^{2}}{4}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{4} x^{2}+\frac {5}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-9\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 11921

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +8 y \left (x^{2}-1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{10}-\frac {x^{6}}{90}+O\left (x^{7}\right )\right )+c_{2} \left (-\frac {4 x^{4} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{10}-\frac {x^{6}}{90}+O\left (x^{7}\right )\right ) \ln \left (x \right )}{3}+\frac {1+x^{2}+x^{4}+O\left (x^{7}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1-\frac {1}{2} x^{2}+\frac {1}{10} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400-86400 x^{2}-86400 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 11922

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{2}-\frac {3 y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x}{2}+\frac {5 x^{2}}{32}-\frac {7 x^{3}}{192}+\frac {7 x^{4}}{1024}-\frac {11 x^{5}}{10240}+O\left (x^{6}\right )\right )+c_{2} \left (\frac {x^{\frac {3}{2}} \left (1-\frac {x}{2}+\frac {5 x^{2}}{32}-\frac {7 x^{3}}{192}+\frac {7 x^{4}}{1024}-\frac {11 x^{5}}{10240}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{8}+\frac {1-\frac {x}{2}+\frac {x^{3}}{24}-\frac {61 x^{4}}{3072}+\frac {59 x^{5}}{10240}+O\left (x^{6}\right )}{\sqrt {x}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{2} x +\frac {5}{32} x^{2}-\frac {7}{192} x^{3}+\frac {7}{1024} x^{4}-\frac {11}{10240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {1}{4} x^{2}+\frac {1}{8} x^{3}-\frac {5}{128} x^{4}+\frac {7}{768} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+x -\frac {1}{12} x^{3}+\frac {61}{1536} x^{4}-\frac {59}{5120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

Problem 11923

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (x^{2}-2 x +1-\frac {2 x^{3}}{9}+\frac {x^{4}}{36}-\frac {x^{5}}{450}+O\left (x^{6}\right )\right )+c_{2} \left (\left (x^{2}-2 x +1-\frac {2 x^{3}}{9}+\frac {x^{4}}{36}-\frac {x^{5}}{450}+O\left (x^{6}\right )\right ) \ln \left (x \right )-3 x^{2}+4 x +\frac {22 x^{3}}{27}-\frac {25 x^{4}}{216}+\frac {137 x^{5}}{13500}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-2 x +x^{2}-\frac {2}{9} x^{3}+\frac {1}{36} x^{4}-\frac {1}{450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (4 x -3 x^{2}+\frac {22}{27} x^{3}-\frac {25}{216} x^{4}+\frac {137}{13500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 11924

ODE	\[ \boxed {2 x y^{\prime \prime }+6 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x}{6}+\frac {x^{2}}{96}-\frac {x^{3}}{2880}+\frac {x^{4}}{138240}-\frac {x^{5}}{9676800}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-\frac {1}{8}+\frac {x}{48}-\frac {x^{2}}{768}+\frac {x^{3}}{23040}-\frac {x^{4}}{1105920}+\frac {x^{5}}{77414400}-\frac {O\left (x^{6}\right )}{8}\right ) \ln \left (x \right )+\frac {1+\frac {x}{2}-\frac {x^{3}}{36}+\frac {25 x^{4}}{9216}-\frac {157 x^{5}}{1382400}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{6} x +\frac {1}{96} x^{2}-\frac {1}{2880} x^{3}+\frac {1}{138240} x^{4}-\frac {1}{9676800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{384} x^{4}-\frac {1}{11520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-x +\frac {1}{18} x^{3}-\frac {25}{4608} x^{4}+\frac {157}{691200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

Problem 11925

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (\frac {x^{2}}{4}-\frac {3 x^{4}}{128}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x \]

Problem 11926

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +\left (x^{2}-3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {x^{2}}{12}+\frac {x^{4}}{384}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{3} \left (1-\frac {x^{2}}{12}+\frac {x^{4}}{384}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{16}+\frac {1+\frac {x^{2}}{4}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 11927

ODE	\begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=2 x \left (t \right )+4 y \left (t \right )+{\mathrm e}^{t}\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=y \left (t \right )+{\mathrm e}^{4 t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= \frac {{\mathrm e}^{4 t}}{3}-\frac {{\mathrm e}^{t}}{3}-\frac {2 c_{1} {\mathrm e}^{-2 t}}{3} \\ \end{align*}

Problem 11928

ODE	\begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )-2 t\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=t^{2}+3 x \left (t \right )+y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -2+{\mathrm e}^{-t} c_{1} \\ y \left (t \right ) &= -t^{2}+4-2 \,{\mathrm e}^{-t} c_{1} -2 t \\ \end{align*}

Problem 11929

ODE	\begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )+3 y \left (t \right )+{\mathrm e}^{t}\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=-x \left (t \right )+{\mathrm e}^{3 t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{t}}{4}+c_{1} {\mathrm e}^{-3 t} \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t}}{3}-\frac {{\mathrm e}^{t}}{2}-\frac {2 c_{1} {\mathrm e}^{-3 t}}{3} \\ \end{align*}

Problem 11930

ODE	\begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )+2 \,{\mathrm e}^{t}\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=3 x \left (t \right )+4 y \left (t \right )+{\mathrm e}^{2 t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{t} \\ y \left (t \right ) &= -\frac {{\mathrm e}^{2 t}}{2}-2 \,{\mathrm e}^{t} \\ \end{align*}

Problem 11931

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+2 y \left (t \right )-{\mathrm e}^{t}+{\mathrm e}^{-t}\\ y^{\prime }\left (t \right )&=-5 x \left (t \right )-3 y \left (t \right )+2 \,{\mathrm e}^{t}-{\mathrm e}^{-t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ y \left (t \right ) &= \frac {c_{1} \cos \left (t \right )}{2}-\frac {3 c_{2} \cos \left (t \right )}{2}-\frac {3 c_{1} \sin \left (t \right )}{2}-\frac {c_{2} \sin \left (t \right )}{2}+\frac {{\mathrm e}^{t}}{2}-\frac {{\mathrm e}^{-t}}{2} \\ \end{align*}

Problem 11932

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+t -{\mathrm e}^{t}\\ y^{\prime }\left (t \right )&=5 x \left (t \right )+y \left (t \right )-t +2 \,{\mathrm e}^{t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= t -1-\frac {{\mathrm e}^{t}}{2}+c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= -\frac {5 c_{2} {\mathrm e}^{-t}}{2}-4 t +1+c_{1} {\mathrm e}^{t}-\frac {{\mathrm e}^{t} t}{2} \\ \end{align*}

Problem 11933

ODE	\begin {align*} x^{\prime }\left (t \right )&=-t +2 \,{\mathrm e}^{3 t}+6 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+t -{\mathrm e}^{3 t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{\sqrt {6}\, t} c_{2} +{\mathrm e}^{-\sqrt {6}\, t} c_{1} -t +\frac {1}{6} \\ y \left (t \right ) &= \frac {\sqrt {6}\, {\mathrm e}^{\sqrt {6}\, t} c_{2}}{6}-\frac {\sqrt {6}\, {\mathrm e}^{-\sqrt {6}\, t} c_{1}}{6}-\frac {1}{6}+\frac {t}{6}-\frac {{\mathrm e}^{3 t}}{3} \\ \end{align*}

Problem 11934

ODE	\begin {align*} x^{\prime }\left (t \right )&=6 t -1+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-3 t +1 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{\sqrt {3}\, t} c_{2} +{\mathrm e}^{-\sqrt {3}\, t} c_{1} +3 t -3 \\ y \left (t \right ) &= \frac {\sqrt {3}\, {\mathrm e}^{\sqrt {3}\, t} c_{2}}{3}-\frac {\sqrt {3}\, {\mathrm e}^{-\sqrt {3}\, t} c_{1}}{3}+\frac {4}{3}-2 t \\ \end{align*}

Problem 11935

ODE	\begin {align*} x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=-2 y \left (t \right )+\sin \left (t \right )\\ x^{\prime }\left (t \right )+y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}-\frac {\sin \left (t \right )}{2} \\ y \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{t}}{3}+\frac {\sin \left (t \right )}{2} \\ \end{align*}

Problem 11936

ODE	\begin {align*} x^{\prime }\left (t \right )&=\frac {3 x \left (t \right )}{2}-\frac {3 y \left (t \right )}{2}+\frac {t}{2}+\frac {1}{2}\\ y^{\prime }\left (t \right )&=-\frac {x \left (t \right )}{2}+\frac {5 y \left (t \right )}{2}-\frac {t}{2}+\frac {1}{2} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{1} {\mathrm e}^{3 t}-\frac {t}{6}-\frac {13}{18} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{t}}{3}-c_{1} {\mathrm e}^{3 t}-\frac {5}{18}+\frac {t}{6} \\ \end{align*}

Problem 11937

ODE	\begin {align*} x^{\prime }\left (t \right )&=-2+x \left (t \right )-3 y \left (t \right )+4 t\\ y^{\prime }\left (t \right )&=4-3 x \left (t \right )+y \left (t \right )-4 t \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{4 t}+c_{1} {\mathrm e}^{-2 t}-t +1 \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{4 t}+c_{1} {\mathrm e}^{-2 t}+t \\ \end{align*}

Problem 11938

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 t^{2}-6 y \left (t \right )-2 t -1\\ y^{\prime }\left (t \right )&=-t^{2}+x \left (t \right )+y \left (t \right )+2 t +1 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_{2} +{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_{1} +\frac {2 t^{2}}{3}-\frac {7 t}{9}-\frac {41}{27} \\ y \left (t \right ) &= \frac {t^{2}}{3}-\frac {{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_{2}}{12}-\frac {{\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_{2}}{12}-\frac {{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_{1}}{12}+\frac {{\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_{1}}{12}-\frac {5 t}{9}-\frac {1}{27} \\ \end{align*}

Problem 11939

ODE	\begin {align*} x^{\prime }\left (t \right )&=-t^{2}+x \left (t \right )+y \left (t \right )+6 t\\ y^{\prime }\left (t \right )&=3 t^{2}-3 x \left (t \right )-3 y \left (t \right )-8 t \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -\frac {c_{1} {\mathrm e}^{-2 t}}{2}+2 t^{2}+t +c_{2} \\ y \left (t \right ) &= -t^{2}+\frac {3 c_{1} {\mathrm e}^{-2 t}}{2}-3 t +1-c_{2} \\ \end{align*}

Problem 11940

ODE	\begin {align*} x^{\prime }\left (t \right )&=-5-x \left (t \right )-t -y \left (t \right )\\ y^{\prime }\left (t \right )&=7+2 x \left (t \right )+2 t +y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+c_{1} \cos \left (t \right )-3-t \\ y \left (t \right ) &= -c_{2} \cos \left (t \right )+c_{1} \sin \left (t \right )-1-c_{2} \sin \left (t \right )-c_{1} \cos \left (t \right ) \\ \end{align*}

Problem 11941

ODE	\begin {align*} x^{\prime }\left (t \right )&=-\frac {7 x \left (t \right )}{2}-\frac {9 y \left (t \right )}{2}+\frac {{\mathrm e}^{t}}{2}\\ y^{\prime }\left (t \right )&=\frac {3 x \left (t \right )}{2}+\frac {5 y \left (t \right )}{2}+\frac {{\mathrm e}^{t}}{2} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{1} {\mathrm e}^{-2 t}-{\mathrm e}^{t} t \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{t}-\frac {c_{1} {\mathrm e}^{-2 t}}{3}+{\mathrm e}^{t} t +\frac {{\mathrm e}^{t}}{3} \\ \end{align*}

Problem 11942

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-2 t -t^{2}\\ y^{\prime }\left (t \right )&=-3 x \left (t \right )+y \left (t \right )+2 t +2 t^{2} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {t^{2}}{2}+\frac {3 t}{2}+\frac {3}{4}+c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= \frac {15}{4}-3 c_{2} {\mathrm e}^{2 t}+\frac {3 t}{2}-\frac {t^{2}}{2}+c_{1} {\mathrm e}^{t} \\ \end{align*}

Problem 11943

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-t +1\\ y^{\prime }\left (t \right )&=-5 x \left (t \right )+y \left (t \right )+2 t -1 \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= \frac {t}{3}-\frac {2}{9}+c_{2} {\mathrm e}^{3 t} \\ y \left (t \right ) &= -\frac {4}{9}-\frac {5 c_{2} {\mathrm e}^{3 t}}{2}-\frac {t}{3}+c_{1} {\mathrm e}^{t} \\ \end{align*}

Problem 11944

ODE	\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+4 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 2] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= 2 \,{\mathrm e}^{5 t}-{\mathrm e}^{-t} \\ y \left (t \right ) &= {\mathrm e}^{5 t}+{\mathrm e}^{-t} \\ \end{align*}

Problem 11945

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=4 x \left (t \right )+y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 8] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{7 t}-3 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{7 t}+6 \,{\mathrm e}^{-t} \\ \end{align*}

Problem 11946

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+2 y \left (t \right )+5 t\\ y^{\prime }\left (t \right )&=3 x \left (t \right )+4 y \left (t \right )+17 t \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{7 t}+c_{1} {\mathrm e}^{2 t}+t +1 \\ y \left (t \right ) &= c_{2} {\mathrm e}^{7 t}-\frac {3 c_{1} {\mathrm e}^{2 t}}{2}-2-5 t \\ \end{align*}

Problem 11947

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{3 t} \\ y \left (t \right ) &= 2 c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{3 t} \\ \end{align*}

Problem 11948

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \left (t \right )+y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{4 t}+3 c_{2} {\mathrm e}^{2 t} \\ \end{align*}

Problem 11949

ODE	\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+7 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \left (t \right )+2 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 9, y \left (0\right ) = -1] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= 2 \,{\mathrm e}^{5 t}+7 \,{\mathrm e}^{-5 t} \\ y \left (t \right ) &= 2 \,{\mathrm e}^{5 t}-3 \,{\mathrm e}^{-5 t} \\ \end{align*}

Problem 11950

ODE	\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=7 x \left (t \right )+4 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 6, y \left (0\right ) = 2] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{5 t}+5 \,{\mathrm e}^{-3 t} \\ y \left (t \right ) &= 7 \,{\mathrm e}^{5 t}-5 \,{\mathrm e}^{-3 t} \\ \end{align*}

Problem 11965

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+3 y \left (t \right )-4 z \left (t \right )\\ z^{\prime }\left (t \right )&=4 x \left (t \right )+y \left (t \right )-4 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{-3 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+2 c_{2} {\mathrm e}^{2 t}+7 c_{3} {\mathrm e}^{-3 t} \\ z \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t}+11 c_{3} {\mathrm e}^{-3 t} \\ \end{align*}

Problem 11966

ODE	\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )-z \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+3 y \left (t \right )+z \left (t \right )\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )-6 y \left (t \right )+6 z \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{2 t}+c_{3} {\mathrm e}^{5 t} \\ y \left (t \right ) &= -c_{1} {\mathrm e}^{3 t}-\frac {7 c_{2} {\mathrm e}^{2 t}}{10}-c_{3} {\mathrm e}^{5 t} \\ z \left (t \right ) &= -c_{1} {\mathrm e}^{3 t}-\frac {3 c_{2} {\mathrm e}^{2 t}}{10}-3 c_{3} {\mathrm e}^{5 t} \\ \end{align*}

Problem 11967

ODE	\[ \boxed {x^{\prime }=\cos \left (t \right )+\sin \left (t \right )} \]
program solution	\[ x = \sin \left (t \right )-\cos \left (t \right )+c_{1} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\cos \left (t \right )+\sin \left (t \right )+c_{1} \]

Problem 11968

ODE	\[ \boxed {y^{\prime }=\frac {1}{x^{2}-1}} \]
program solution	\[ y = -\operatorname {arctanh}\left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\operatorname {arctanh}\left (x \right )+c_{1} \]

Problem 11969

ODE	\[ \boxed {u^{\prime }=4 t \ln \left (t \right )} \]
program solution	\[ u = 2 t^{2} \ln \left (t \right )-t^{2}+c_{1} \] Verified OK.
Maple solution	\[ u \left (t \right ) = 2 \ln \left (t \right ) t^{2}-t^{2}+c_{1} \]

Problem 11970

ODE	\[ \boxed {z^{\prime }=x \,{\mathrm e}^{-2 x}} \]
program solution	\[ z = \left (-\frac {x}{2}-\frac {1}{4}\right ) {\mathrm e}^{-2 x}+c_{1} \] Verified OK.
Maple solution	\[ z \left (x \right ) = \frac {\left (-2 x -1\right ) {\mathrm e}^{-2 x}}{4}+c_{1} \]

Problem 11971

ODE	\[ \boxed {T^{\prime }={\mathrm e}^{-t} \sin \left (2 t \right )} \]
program solution	\[ T = -\frac {\left (\sin \left (2 t \right )+2 \cos \left (2 t \right )\right ) {\mathrm e}^{-t}}{5}+c_{1} \] Verified OK.
Maple solution	\[ T \left (t \right ) = \frac {{\mathrm e}^{-t} \left (-2 \cos \left (2 t \right )-\sin \left (2 t \right )\right )}{5}+c_{1} \]

Problem 11972

ODE	\[ \boxed {x^{\prime }=\sec \left (t \right )^{2}} \] With initial conditions \begin {align*} \left [x \left (\frac {\pi }{4}\right ) = 0\right ] \end {align*}
program solution	\[ x = \tan \left (t \right )-1 \] Verified OK.
Maple solution	\[ x \left (t \right ) = \tan \left (t \right )-1 \]

Problem 11973

ODE	\[ \boxed {y^{\prime }=x -\frac {1}{3} x^{3}} \] With initial conditions \begin {align*} [y \left (-1\right ) = 1] \end {align*}
program solution	\[ y = -\frac {1}{12} x^{4}+\frac {1}{2} x^{2}+\frac {7}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (x^{2}-3\right )^{2}}{12}+\frac {4}{3} \]

Problem 11974

ODE	\[ \boxed {x^{\prime }=2 \sin \left (t \right )^{2}} \] With initial conditions \begin {align*} \left [x \left (\frac {\pi }{4}\right ) = \frac {\pi }{4}\right ] \end {align*}
program solution	\[ x = t -\frac {\sin \left (2 t \right )}{2}+\frac {1}{2} \] Verified OK.
Maple solution	\[ x \left (t \right ) = t +\frac {1}{2}-\frac {\sin \left (2 t \right )}{2} \]

Problem 11975

ODE	\[ \boxed {x V^{\prime }=x^{2}+1} \] With initial conditions \begin {align*} [V \left (1\right ) = 1] \end {align*}
program solution	\[ V = \frac {x^{2}}{2}+\ln \left (x \right )+\frac {1}{2} \] Verified OK.
Maple solution	\[ V \left (x \right ) = \frac {x^{2}}{2}+\ln \left (x \right )+\frac {1}{2} \]

Problem 11976

ODE	\[ \boxed {x^{\prime } {\mathrm e}^{3 t}+3 x \,{\mathrm e}^{3 t}={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [x \left (0\right ) = 3] \end {align*}
program solution	\[ x = 4 \,{\mathrm e}^{-3 t}-{\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\left ({\mathrm e}^{-t}-4\right ) {\mathrm e}^{-3 t} \]

Problem 11977

ODE	\[ \boxed {x^{\prime }+x=1} \]
program solution	\[ x = \frac {{\mathrm e}^{-t}}{c_{1}}+1 \] Verified OK.
Maple solution	\[ x \left (t \right ) = 1+{\mathrm e}^{-t} c_{1} \]

Problem 11978

ODE	\[ \boxed {x^{\prime }-x \left (2-x\right )=0} \]
program solution	\[ x = \frac {2 \,{\mathrm e}^{2 t} c_{1}^{2}}{-1+{\mathrm e}^{2 t} c_{1}^{2}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {2}{1+2 \,{\mathrm e}^{-2 t} c_{1}} \]

Problem 11979

ODE	\[ \boxed {x^{\prime }-\left (x+1\right ) \left (2-x\right ) \sin \left (x\right )=0} \]
program solution	\[ \int _{}^{x}-\frac {1}{\sin \left (\textit {\_a} \right ) \left (\textit {\_a} +1\right ) \left (\textit {\_a} -2\right )}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ t +\int _{}^{x \left (t \right )}\frac {\csc \left (\textit {\_a} \right )}{\left (\textit {\_a} +1\right ) \left (\textit {\_a} -2\right )}d \textit {\_a} +c_{1} = 0 \]

Problem 11980

ODE	\[ \boxed {x^{\prime }+x \left (-x+1\right ) \left (2-x\right )=0} \]
program solution	\[ x = \frac {{\mathrm e}^{t} c_{1}}{\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}}+1 \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {c_{1} {\mathrm e}^{t}+\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}}{\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}} \]

Problem 11981

ODE	\[ \boxed {x^{\prime }-x^{2}+x^{4}=0} \]
program solution	\[ \int _{}^{x}\frac {1}{-\textit {\_a}^{4}+\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right ) {\mathrm e}^{\textit {\_Z}}+2 c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 t \,{\mathrm e}^{\textit {\_Z}}-\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right )-2 c_{1} +\textit {\_Z} -2 t +2\right )}-1 \]

Problem 11982

ODE	\[ \boxed {x^{\prime }-t^{3} \left (-x+1\right )=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 3] \end {align*}
program solution	\[ x = 1+2 \,{\mathrm e}^{-\frac {t^{4}}{4}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = 1+2 \,{\mathrm e}^{-\frac {t^{4}}{4}} \]

Problem 11983

ODE	\[ \boxed {y^{\prime }-\left (1+y^{2}\right ) \tan \left (x \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {i \cos \left (x \right )^{2 i}+1}{\cos \left (x \right )^{2 i}+i} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \cot \left (\frac {\pi }{4}+\ln \left (\cos \left (x \right )\right )\right ) \]

Problem 11984

ODE	\[ \boxed {x^{\prime }-x t^{2}=0} \]
program solution	\[ x = {\mathrm e}^{\frac {t^{3}}{3}+c_{1}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = c_{1} {\mathrm e}^{\frac {t^{3}}{3}} \]

Problem 11985

ODE	\[ \boxed {x^{\prime }+x^{2}=0} \]
program solution	\[ x = \frac {1}{t +c_{1}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {1}{t +c_{1}} \]

Problem 11986

ODE	\[ \boxed {y^{\prime }-y^{2} {\mathrm e}^{-t^{2}}=0} \]
program solution	\[ y = -\frac {2}{\sqrt {\pi }\, \left (c_{3} +\operatorname {erf}\left (t \right )\right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {2}{\sqrt {\pi }\, \operatorname {erf}\left (t \right )-2 c_{1}} \]

Problem 11987

ODE	\[ \boxed {x^{\prime }+p x=q} \]
program solution	\[ x = -\frac {\frac {{\mathrm e}^{-t p}}{c_{1}}-q}{p} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {{\mathrm e}^{-p t} c_{1} p +q}{p} \]

Problem 11988

ODE	\[ \boxed {y^{\prime } x -y k=0} \]
program solution	\[ y = {\mathrm e}^{k \ln \left (x \right )+c_{1} k} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{k} \]

Problem 11989

ODE	\[ \boxed {i^{\prime }-p \left (t \right ) i=0} \]
program solution	\[ i = {\mathrm e}^{-\left (\int _{}^{t}-p \left (\textit {\_a} \right )d \textit {\_a} \right )+c_{1}} \] Verified OK.
Maple solution	\[ i \left (t \right ) = c_{1} {\mathrm e}^{\int p \left (t \right )d t} \]

Problem 11990

ODE	\[ \boxed {x^{\prime }-\lambda x=0} \]
program solution	\[ x = c_{1} {\mathrm e}^{t \lambda } \] Verified OK.
Maple solution	\[ x \left (t \right ) = c_{1} {\mathrm e}^{\lambda t} \]

Problem 11991

ODE	\[ \boxed {m v^{\prime }-k v^{2}=-m g} \]
program solution	\[ v = -\frac {\tanh \left (\frac {\sqrt {m g k}\, \left (t +c_{1} \right )}{m}\right ) \sqrt {m g k}}{k} \] Verified OK.
Maple solution	\[ v \left (t \right ) = -\frac {\tanh \left (\frac {\sqrt {m g k}\, \left (t +c_{1} \right )}{m}\right ) \sqrt {m g k}}{k} \]

Problem 11992

ODE	\[ \boxed {x^{\prime }-k x+x^{2}=0} \] With initial conditions \begin {align*} [x \left (0\right ) = x_{0}] \end {align*}
program solution	\[ \frac {\ln \left (x\right )-\ln \left (x-k \right )}{k} = t +\frac {\ln \left (x_{0} \right )-\ln \left (-k +x_{0} \right )}{k} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {k x_{0}}{\left (-x_{0} +k \right ) {\mathrm e}^{-k t}+x_{0}} \]

Problem 11993

ODE	\[ \boxed {x^{\prime }+x \left (k^{2}+x^{2}\right )=0} \] With initial conditions \begin {align*} [x \left (0\right ) = x_{0}] \end {align*}
program solution	\[ \frac {-2 \ln \left (x\right )+\ln \left (k^{2}+x^{2}\right )}{2 k^{2}} = t +\frac {-2 \ln \left (x_{0} \right )+\ln \left (k^{2}+x_{0}^{2}\right )}{2 k^{2}} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 11994

ODE	\[ \boxed {y^{\prime }+\frac {y}{x}=x^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = y_{0}] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 11995

ODE	\[ \boxed {x t +x^{\prime }=4 t} \] With initial conditions \begin {align*} [x \left (0\right ) = 2] \end {align*}
program solution	\[ x = 4-2 \,{\mathrm e}^{-\frac {t^{2}}{2}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = 4-2 \,{\mathrm e}^{-\frac {t^{2}}{2}} \]

Problem 11996

ODE	\[ \boxed {z^{\prime }-z \tan \left (y \right )=\sin \left (y \right )} \]
program solution	\[ z = -\frac {\cos \left (y \right )^{2}-2 c_{1}}{2 \cos \left (y \right )} \] Verified OK.
Maple solution	\[ z \left (y \right ) = -\frac {\cos \left (y \right )}{2}+\sec \left (y \right ) c_{1} +\frac {\sec \left (y \right )}{4} \]

Problem 11997

ODE	\[ \boxed {y^{\prime }+{\mathrm e}^{-x} y=1} \] With initial conditions \begin {align*} [y \left (0\right ) = {\mathrm e}] \end {align*}
program solution	\[ y = {\mathrm e}^{{\mathrm e}^{-x}} \operatorname {expIntegral}_{1}\left ({\mathrm e}^{-x}\right )-{\mathrm e}^{{\mathrm e}^{-x}} \operatorname {expIntegral}_{1}\left (1\right )+{\mathrm e}^{{\mathrm e}^{-x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-x}\right )+1-\operatorname {expIntegral}_{1}\left (1\right )\right ) {\mathrm e}^{{\mathrm e}^{-x}} \]

Problem 11998

ODE	\[ \boxed {x^{\prime }+x \tanh \left (t \right )=3} \]
program solution	\[ x = \frac {3 \sinh \left (t \right )+c_{1}}{\cosh \left (t \right )} \] Verified OK.
Maple solution	\[ x \left (t \right ) = 3 \tanh \left (t \right )+\operatorname {sech}\left (t \right ) c_{1} \]

Problem 11999

ODE	\[ \boxed {y^{\prime }+2 y \cot \left (x \right )=5} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = 1\right ] \end {align*}
program solution	\[ y = -\frac {5 \csc \left (x \right )^{2} \sin \left (x \right ) \cos \left (x \right )}{2}-\frac {5 \csc \left (x \right )^{2} \pi }{4}+\frac {5 \csc \left (x \right )^{2} x}{2}+\csc \left (x \right )^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-10 x +5 \sin \left (2 x \right )-4+5 \pi }{-2+2 \cos \left (2 x \right )} \]

Problem 12000

ODE	\[ \boxed {x^{\prime }+5 x=t} \]
program solution	\[ x = \frac {\left (5 t \,{\mathrm e}^{5 t}-{\mathrm e}^{5 t}+25 c_{1} \right ) {\mathrm e}^{-5 t}}{25} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {t}{5}-\frac {1}{25}+{\mathrm e}^{-5 t} c_{1} \]