2.17.149 Problems 14801 to 14900

Problem 14801

ODE	\[ \boxed {x^{2} y^{\prime \prime }+6 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{2}+\frac {i \sqrt {23}}{2}} \left (1+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{2}-\frac {i \sqrt {23}}{2}} \left (1+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} x^{-\frac {i \sqrt {23}}{2}}+c_{2} x^{\frac {i \sqrt {23}}{2}}\right )+O\left (x^{6}\right ) \]

Problem 14802

ODE	\[ \boxed {x \left (x +1\right ) y^{\prime \prime }+\frac {y^{\prime }}{x^{2}}+5 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 14803

ODE	\[ \boxed {\left (x^{2}-3 x -4\right ) y^{\prime \prime }-\left (x +1\right ) y^{\prime }+y \left (x^{2}-1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{8} x^{2}+\frac {1}{24} x^{3}+\frac {1}{192} x^{4}-\frac {1}{640} x^{5}+\frac {1}{9216} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{8} x^{2}-\frac {1}{24} x^{3}+\frac {1}{48} x^{4}+\frac {1}{960} x^{5}-\frac {1}{2560} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{8} x^{2}+\frac {1}{24} x^{3}+\frac {1}{192} x^{4}-\frac {1}{640} x^{5}\right ) c_{1} +\left (x -\frac {1}{8} x^{2}-\frac {1}{24} x^{3}+\frac {1}{48} x^{4}+\frac {1}{960} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{8} x^{2}+\frac {1}{24} x^{3}+\frac {1}{192} x^{4}-\frac {1}{640} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{8} x^{2}-\frac {1}{24} x^{3}+\frac {1}{48} x^{4}+\frac {1}{960} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14804

ODE	\[ \boxed {\left (x^{2}-25\right )^{2} y^{\prime \prime }-\left (x +5\right ) y^{\prime }+10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{125} x^{2}-\frac {1}{46875} x^{3}-\frac {767}{7812500} x^{4}-\frac {4813}{7324218750} x^{5}-\frac {6201119}{2746582031250} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{250} x^{2}-\frac {112}{46875} x^{3}+\frac {173}{3906250} x^{4}-\frac {409681}{7324218750} x^{5}+\frac {4953719}{5493164062500} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{125} x^{2}-\frac {1}{46875} x^{3}-\frac {767}{7812500} x^{4}-\frac {4813}{7324218750} x^{5}\right ) c_{1} +\left (x +\frac {1}{250} x^{2}-\frac {112}{46875} x^{3}+\frac {173}{3906250} x^{4}-\frac {409681}{7324218750} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{125} x^{2}-\frac {1}{46875} x^{3}-\frac {767}{7812500} x^{4}-\frac {4813}{7324218750} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{250} x^{2}-\frac {112}{46875} x^{3}+\frac {173}{3906250} x^{4}-\frac {409681}{7324218750} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14805

ODE	\[ \boxed {2 x y^{\prime \prime }-5 y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{2}} \left (1+\frac {x}{3}+\frac {x^{2}}{22}+\frac {x^{3}}{286}+\frac {x^{4}}{5720}+\frac {3 x^{5}}{486200}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {3 x}{5}+\frac {3 x^{2}}{10}-\frac {3 x^{3}}{10}-\frac {9 x^{4}}{40}-\frac {9 x^{5}}{200}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {7}{2}} \left (1+\frac {1}{3} x +\frac {1}{22} x^{2}+\frac {1}{286} x^{3}+\frac {1}{5720} x^{4}+\frac {3}{486200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {3}{5} x +\frac {3}{10} x^{2}-\frac {3}{10} x^{3}-\frac {9}{40} x^{4}-\frac {9}{200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 14806

ODE	\[ \boxed {5 x y^{\prime \prime }+8 y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{26}+\frac {x^{4}}{2392}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x^{2}}{14}+\frac {x^{4}}{952}+O\left (x^{6}\right )\right )}{x^{\frac {3}{5}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{14} x^{2}+\frac {1}{952} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{5}}}+c_{2} \left (1+\frac {1}{26} x^{2}+\frac {1}{2392} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 14807

ODE	\[ \boxed {9 x y^{\prime \prime }+14 y^{\prime }+y \left (x -1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x}{14}-\frac {13 x^{2}}{644}-\frac {59 x^{3}}{61824}+\frac {29 x^{4}}{247296}+\frac {53 x^{5}}{12364800}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{4}-\frac {3 x^{2}}{104}-\frac {29 x^{3}}{6864}+\frac {13 x^{4}}{65472}+\frac {251 x^{5}}{11348480}+O\left (x^{6}\right )\right )}{x^{\frac {5}{9}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{4} x -\frac {3}{104} x^{2}-\frac {29}{6864} x^{3}+\frac {13}{65472} x^{4}+\frac {251}{11348480} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {5}{9}}}+c_{2} \left (1+\frac {1}{14} x -\frac {13}{644} x^{2}-\frac {59}{61824} x^{3}+\frac {29}{247296} x^{4}+\frac {53}{12364800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 14808

ODE	\[ \boxed {7 x y^{\prime \prime }+10 y^{\prime }+\left (-x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x}{10}+\frac {x^{2}}{340}+\frac {113 x^{3}}{8160}-\frac {929 x^{4}}{1011840}+\frac {781 x^{5}}{38449920}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x}{4}+\frac {x^{2}}{88}+\frac {29 x^{3}}{1584}-\frac {17 x^{4}}{6336}+\frac {89 x^{5}}{1013760}+O\left (x^{6}\right )\right )}{x^{\frac {3}{7}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{4} x +\frac {1}{88} x^{2}+\frac {29}{1584} x^{3}-\frac {17}{6336} x^{4}+\frac {89}{1013760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{7}}}+c_{2} \left (1-\frac {1}{10} x +\frac {1}{340} x^{2}+\frac {113}{8160} x^{3}-\frac {929}{1011840} x^{4}+\frac {781}{38449920} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 14809

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 14810

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	\[ y = c_{1} x \left (1-\frac {3 x}{2}+\frac {5 x^{2}}{4}-\frac {35 x^{3}}{48}+\frac {21 x^{4}}{64}-\frac {77 x^{5}}{640}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-\frac {3 x}{2}+\frac {5 x^{2}}{4}-\frac {35 x^{3}}{48}+\frac {21 x^{4}}{64}-\frac {77 x^{5}}{640}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {5 x^{2}}{4}+\frac {19 x^{3}}{12}-\frac {73 x^{4}}{64}+\frac {1129 x^{5}}{1920}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {3}{2} x +\frac {5}{4} x^{2}-\frac {35}{48} x^{3}+\frac {21}{64} x^{4}-\frac {77}{640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {3}{2} x^{2}-\frac {5}{4} x^{3}+\frac {35}{48} x^{4}-\frac {21}{64} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-2 x +\frac {7}{4} x^{2}-\frac {11}{12} x^{3}+\frac {61}{192} x^{4}-\frac {131}{1920} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 14811

ODE	\[ \boxed {y^{\prime \prime }+\frac {8 y^{\prime }}{3 x}-\left (\frac {2}{3 x^{2}}-1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1-\frac {3 x^{2}}{26}+\frac {9 x^{4}}{1976}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {3 x^{2}}{2}-\frac {9 x^{4}}{40}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {7}{3}} \left (1-\frac {3}{26} x^{2}+\frac {9}{1976} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+\frac {3}{2} x^{2}-\frac {9}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 14812

ODE	\[ \boxed {y^{\prime \prime }+\left (\frac {16}{3 x}-1\right ) y^{\prime }-\frac {16 y}{3 x^{2}}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {3 x}{22}+\frac {9 x^{2}}{550}+\frac {27 x^{3}}{15400}+\frac {81 x^{4}}{477400}+\frac {243 x^{5}}{16231600}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right )}{x^{\frac {16}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {16}{3}}}+c_{2} x \left (1+\frac {3}{22} x +\frac {9}{550} x^{2}+\frac {27}{15400} x^{3}+\frac {81}{477400} x^{4}+\frac {243}{16231600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 14813

ODE	\[ \boxed {y^{\prime \prime }+\left (\frac {1}{2 x}-2\right ) y^{\prime }-\frac {35 y}{16 x^{2}}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{4}} \left (1+\frac {7 x}{8}+\frac {77 x^{2}}{160}+\frac {77 x^{3}}{384}+\frac {209 x^{4}}{3072}+\frac {4807 x^{5}}{245760}+O\left (x^{6}\right )\right )+c_{2} \left (\frac {5 x^{\frac {7}{4}} \left (1+\frac {7 x}{8}+\frac {77 x^{2}}{160}+\frac {77 x^{3}}{384}+\frac {209 x^{4}}{3072}+\frac {4807 x^{5}}{245760}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{32}+\frac {1+\frac {5 x}{4}+\frac {5 x^{2}}{16}-\frac {95 x^{4}}{1024}-\frac {1563 x^{5}}{20480}+O\left (x^{6}\right )}{x^{\frac {5}{4}}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{3} \left (1+\frac {7}{8} x +\frac {77}{160} x^{2}+\frac {77}{384} x^{3}+\frac {209}{3072} x^{4}+\frac {4807}{245760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {15}{8} x^{3}+\frac {105}{64} x^{4}+\frac {231}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+15 x +\frac {15}{4} x^{2}-\frac {13}{2} x^{3}-\frac {1741}{256} x^{4}-\frac {4141}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{\frac {5}{4}}} \]

Problem 14814

ODE	\[ \boxed {y^{\prime \prime }-\left (\frac {1}{x}+2\right ) y^{\prime }+\left (x +\frac {1}{x^{2}}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (2 x^{2}+2 x +1+\frac {11 x^{3}}{9}+\frac {35 x^{4}}{72}+\frac {103 x^{5}}{900}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (2 x^{2}+2 x +1+\frac {11 x^{3}}{9}+\frac {35 x^{4}}{72}+\frac {103 x^{5}}{900}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (-3 x^{2}-2 x -\frac {64 x^{3}}{27}-\frac {497 x^{4}}{432}-\frac {9371 x^{5}}{27000}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+2 x +2 x^{2}+\frac {11}{9} x^{3}+\frac {35}{72} x^{4}+\frac {103}{900} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -3 x^{2}-\frac {64}{27} x^{3}-\frac {497}{432} x^{4}-\frac {9371}{27000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x \]

Problem 14815

ODE	\[ \boxed {x^{2} y^{\prime \prime }+7 y^{\prime } x -7 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+O\left (x^{9}\right )\right )+\frac {c_{2} \left (1+O\left (x^{9}\right )\right )}{x^{7}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-203212800+\operatorname {O}\left (x^{6}\right )\right )}{x^{7}} \]

Problem 14816

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

Problem 14817

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

Problem 14818

ODE	\[ \boxed {y^{\prime \prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{3}}{6}\right ) c_{1} +\left (x -\frac {1}{12} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14819

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

Problem 14820

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +k \left (k +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2} k^{2}-\frac {1}{2} x^{2} k +\frac {1}{24} k^{4} x^{4}+\frac {1}{12} k^{3} x^{4}-\frac {5}{24} k^{2} x^{4}-\frac {1}{4} k \,x^{4}-\frac {1}{720} x^{6} k^{6}-\frac {1}{240} x^{6} k^{5}+\frac {23}{720} x^{6} k^{4}+\frac {17}{240} x^{6} k^{3}-\frac {47}{360} x^{6} k^{2}-\frac {1}{6} x^{6} k \right ) y \left (0\right )+\left (x -\frac {1}{6} k^{2} x^{3}-\frac {1}{6} k \,x^{3}+\frac {1}{3} x^{3}+\frac {1}{120} x^{5} k^{4}+\frac {1}{60} x^{5} k^{3}-\frac {13}{120} x^{5} k^{2}-\frac {7}{60} x^{5} k +\frac {1}{5} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\left (-\frac {1}{2} k^{2}-\frac {1}{2} k \right ) x^{2}+\left (-\frac {5}{24} k^{2}-\frac {1}{4} k +\frac {1}{24} k^{4}+\frac {1}{12} k^{3}\right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {1}{6} k^{2}-\frac {1}{6} k +\frac {1}{3}\right ) x^{3}+\left (-\frac {13}{120} k^{2}-\frac {7}{60} k +\frac {1}{5}+\frac {1}{120} k^{4}+\frac {1}{60} k^{3}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {k \left (1+k \right ) x^{2}}{2}+\frac {k \left (k^{3}+2 k^{2}-5 k -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (k^{2}+k -2\right ) x^{3}}{6}+\frac {\left (k^{4}+2 k^{3}-13 k^{2}-14 k +24\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14821

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}-3 x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {3 x}{2}+\frac {15 x^{2}}{8}+\frac {35 x^{3}}{16}+\frac {315 x^{4}}{128}+\frac {693 x^{5}}{256}+O\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {8 x^{2}}{3}+\frac {16 x^{3}}{5}+\frac {128 x^{4}}{35}+\frac {256 x^{5}}{63}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {3}{2} x +\frac {15}{8} x^{2}+\frac {35}{16} x^{3}+\frac {315}{128} x^{4}+\frac {693}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+2 x +\frac {8}{3} x^{2}+\frac {16}{5} x^{3}+\frac {128}{35} x^{4}+\frac {256}{63} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 14822

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

Problem 14823

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

Problem 14824

ODE	\[ \boxed {x y^{\prime \prime }+\left (1-x \right ) y^{\prime }+k y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (-k x +1+\frac {\left (k -1\right ) k \,x^{2}}{4}-\frac {\left (k -2\right ) \left (k -1\right ) k \,x^{3}}{36}+\frac {\left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k \,x^{4}}{576}-\frac {\left (k -4\right ) \left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k \,x^{5}}{14400}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-k x +1+\frac {\left (k -1\right ) k \,x^{2}}{4}-\frac {\left (k -2\right ) \left (k -1\right ) k \,x^{3}}{36}+\frac {\left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k \,x^{4}}{576}-\frac {\left (k -4\right ) \left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k \,x^{5}}{14400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\left (2 k +1\right ) x +\left (-\frac {k}{2}+\frac {1}{4}-\frac {3 \left (k -1\right ) k}{4}\right ) x^{2}+\left (\frac {\left (k -1\right ) k}{36}+\frac {\left (k -2\right ) k}{36}+\frac {\left (k -2\right ) \left (k -1\right )}{36}+\frac {11 \left (k -2\right ) \left (k -1\right ) k}{108}\right ) x^{3}+\left (-\frac {\left (k -2\right ) \left (k -1\right ) k}{576}-\frac {\left (k -3\right ) \left (k -1\right ) k}{576}-\frac {\left (k -3\right ) \left (k -2\right ) k}{576}-\frac {\left (k -3\right ) \left (k -2\right ) \left (k -1\right )}{576}-\frac {25 \left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k}{3456}\right ) x^{4}+\left (\frac {\left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k}{14400}+\frac {\left (k -4\right ) \left (k -2\right ) \left (k -1\right ) k}{14400}+\frac {\left (k -4\right ) \left (k -3\right ) \left (k -1\right ) k}{14400}+\frac {\left (k -4\right ) \left (k -3\right ) \left (k -2\right ) k}{14400}+\frac {\left (k -4\right ) \left (k -3\right ) \left (k -2\right ) \left (k -1\right )}{14400}+\frac {137 \left (k -4\right ) \left (k -3\right ) \left (k -2\right ) \left (k -1\right ) k}{432000}\right ) x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (2 k +1\right ) x +\left (\frac {1}{4} k +\frac {1}{4}-\frac {3}{4} k^{2}\right ) x^{2}+\left (-\frac {2}{9} k^{2}+\frac {1}{27} k +\frac {1}{18}+\frac {11}{108} k^{3}\right ) x^{3}+\left (\frac {7}{192} k^{3}-\frac {167}{3456} k^{2}+\frac {1}{192} k +\frac {1}{96}-\frac {25}{3456} k^{4}\right ) x^{4}+\left (\frac {1}{1500} k -\frac {61}{21600} k^{4}+\frac {1}{600}+\frac {719}{86400} k^{3}-\frac {37}{4320} k^{2}+\frac {137}{432000} k^{5}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1-k x +\frac {1}{4} \left (-1+k \right ) k x^{2}-\frac {1}{36} \left (-2+k \right ) \left (-1+k \right ) k x^{3}+\frac {1}{576} \left (k -3\right ) \left (-2+k \right ) \left (-1+k \right ) k x^{4}-\frac {1}{14400} \left (-4+k \right ) \left (k -3\right ) \left (-2+k \right ) \left (-1+k \right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_{1} +c_{2} \ln \left (x \right )\right ) \]

Problem 14825

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	\[ y \left (x \right ) = \frac {c_{1} x \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 14826

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (16 x^{2}-25\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1-\frac {2 x^{2}}{3}+\frac {4 x^{4}}{21}-\frac {2 x^{6}}{63}+\frac {2 x^{8}}{567}-\frac {4 x^{10}}{14175}+O\left (x^{11}\right )\right )+c_{2} \left (-\frac {32 x^{5} \left (1-\frac {2 x^{2}}{3}+\frac {4 x^{4}}{21}-\frac {2 x^{6}}{63}+\frac {2 x^{8}}{567}-\frac {4 x^{10}}{14175}+O\left (x^{11}\right )\right ) \ln \left (x \right )}{45}+\frac {1+x^{2}+\frac {2 x^{4}}{3}+\frac {4 x^{6}}{9}+\frac {4 x^{8}}{9}+O\left (x^{11}\right )}{x^{5}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{5} \left (1-\frac {2}{3} x^{2}+\frac {4}{21} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-1316818944000-1316818944000 x^{2}-877879296000 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{5}} \]

Problem 14827

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+10 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} {\mathrm e}^{5 x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{5 x}+c_{2} {\mathrm e}^{2 x} \]

Problem 14828

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

Problem 14829

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

Problem 14830

ODE	\[ \boxed {\left (1+t \right )^{2} y^{\prime \prime }-2 y^{\prime } \left (1+t \right )+2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= 1+t \end {align*}
program solution	\[ y = c_{1} \left (1+t \right )+c_{2} t \left (1+t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (t +1\right ) \left (c_{1} +c_{2} \left (t +1\right )\right ) \]

Problem 14831

ODE	\[ \boxed {t y^{\prime \prime }+2 y^{\prime }+y t=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\sin \left (t \right )}{t} \end {align*}
program solution	\[ y = \frac {\sin \left (t \right ) c_{1}}{t}-\frac {c_{2} \sin \left (t \right ) \cot \left (t \right )}{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )}{t} \]

Problem 14832

ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+10 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 x}+\frac {{\mathrm e}^{-2 x} c_{2}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-5 x} c_{1} +{\mathrm e}^{-2 x} c_{2} \]

Problem 14833

ODE	\[ \boxed {6 y^{\prime \prime }+5 y^{\prime }-4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {4 x}{3}}+\frac {6 c_{2} {\mathrm e}^{\frac {x}{2}}}{11} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{\frac {11 x}{6}}+c_{2} \right ) {\mathrm e}^{-\frac {4 x}{3}} \]

Problem 14834

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

Problem 14835

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

Problem 14836

ODE	\[ \boxed {y^{\prime \prime }-10 y^{\prime }+34 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{5 x} \cos \left (3 x \right )+\frac {c_{2} {\mathrm e}^{5 x} \sin \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{5 x} \left (c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right )\right ) \]

Problem 14837

ODE	\[ \boxed {2 y^{\prime \prime }-5 y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x}{2}}+\frac {2 c_{2} {\mathrm e}^{2 x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x}{2}}+c_{2} {\mathrm e}^{2 x} \]

Problem 14838

ODE	\[ \boxed {15 y^{\prime \prime }-11 y^{\prime }+2 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x}{3}}+15 c_{2} {\mathrm e}^{\frac {2 x}{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {x}{3}}+c_{2} {\mathrm e}^{\frac {2 x}{5}} \]

Problem 14839

ODE	\[ \boxed {20 y^{\prime \prime }+y^{\prime }-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{4}}+\frac {20 c_{2} {\mathrm e}^{\frac {x}{5}}}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{\frac {9 x}{20}}+c_{2} \right ) {\mathrm e}^{-\frac {x}{4}} \]

Problem 14840

ODE	\[ \boxed {12 y^{\prime \prime }+8 y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}}+3 c_{2} {\mathrm e}^{-\frac {x}{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x}{2}}+c_{2} {\mathrm e}^{-\frac {x}{6}} \]

Problem 14841

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

Problem 14842

ODE	\[ \boxed {9 y^{\prime \prime \prime }+36 y^{\prime \prime }+40 y^{\prime }=0} \]
program solution	\[ y = c_{1} +{\mathrm e}^{\left (-2+\frac {2 i}{3}\right ) x} c_{2} +{\mathrm e}^{\left (-2-\frac {2 i}{3}\right ) x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +c_{2} {\mathrm e}^{-2 x} \sin \left (\frac {2 x}{3}\right )+c_{3} {\mathrm e}^{-2 x} \cos \left (\frac {2 x}{3}\right ) \]

Problem 14843

ODE	\[ \boxed {9 y^{\prime \prime \prime }+12 y^{\prime \prime }+13 y^{\prime }=0} \]
program solution	\[ y = c_{1} +{\mathrm e}^{\left (-\frac {2}{3}-i\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {2}{3}+i\right ) x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +c_{2} {\mathrm e}^{-\frac {2 x}{3}} \sin \left (x \right )+c_{3} {\mathrm e}^{-\frac {2 x}{3}} \cos \left (x \right ) \]

Problem 14844

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-8 y=-t} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{4 t}}{6}+\frac {t}{8}-\frac {1}{32} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{4 t} c_{2} +{\mathrm e}^{-2 t} c_{1} +\frac {t}{8}-\frac {1}{32} \]

Problem 14845

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }=5 t^{2}} \]
program solution	\[ y = \frac {t^{3}}{3}-\frac {t^{2}}{5}+\frac {c_{1}}{5}+\frac {2 t}{25}-\frac {2}{125}+c_{2} {\mathrm e}^{-5 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {t^{2}}{5}+\frac {t^{3}}{3}-\frac {{\mathrm e}^{-5 t} c_{1}}{5}+\frac {2 t}{25}+c_{2} \]

Problem 14846

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }=-3 \sin \left (t \right )} \]
program solution	\[ y = c_{2} {\mathrm e}^{4 t}-\frac {c_{1}}{4}+\frac {3 \sin \left (t \right )}{17}-\frac {12 \cos \left (t \right )}{17} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} {\mathrm e}^{4 t}}{4}+\frac {3 \sin \left (t \right )}{17}-\frac {12 \cos \left (t \right )}{17}+c_{2} \]

Problem 14847

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=3 \sin \left (2 t \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t} \cos \left (2 t \right )+\frac {c_{2} {\mathrm e}^{-t} \sin \left (2 t \right )}{2}-\frac {12 \cos \left (2 t \right )}{17}+\frac {3 \sin \left (2 t \right )}{17} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sin \left (2 t \right ) \left (17 c_{2} {\mathrm e}^{-t}+3\right )}{17}+{\mathrm e}^{-t} \cos \left (2 t \right ) c_{1} -\frac {12 \cos \left (2 t \right )}{17} \]

Problem 14848

ODE	\[ \boxed {y^{\prime \prime }-9 y=\frac {1}{1+{\mathrm e}^{3 t}}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 t}+\frac {c_{2} {\mathrm e}^{3 t}}{6}-\frac {\left (3 \,{\mathrm e}^{6 t} \ln \left ({\mathrm e}^{t}\right )-{\mathrm e}^{6 t} \ln \left ({\mathrm e}^{t}+1\right )-{\mathrm e}^{6 t} \ln \left ({\mathrm e}^{2 t}-{\mathrm e}^{t}+1\right )+{\mathrm e}^{3 t}+\ln \left (1+{\mathrm e}^{3 t}\right )\right ) {\mathrm e}^{-3 t}}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\ln \left ({\mathrm e}^{2 t}-{\mathrm e}^{t}+1\right ) {\mathrm e}^{6 t}-\ln \left (1+{\mathrm e}^{3 t}\right )+\ln \left (1+{\mathrm e}^{t}\right ) {\mathrm e}^{6 t}+\left (18 c_{1} -3 \ln \left ({\mathrm e}^{t}\right )\right ) {\mathrm e}^{6 t}+18 c_{2} -{\mathrm e}^{3 t}\right ) {\mathrm e}^{-3 t}}{18} \]

Problem 14849

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

Problem 14850

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=-4 \,{\mathrm e}^{-2 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{2 t}-\frac {{\mathrm e}^{-2 t}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (3 c_{1} {\mathrm e}^{4 t}+3 c_{2} {\mathrm e}^{3 t}-1\right ) {\mathrm e}^{-2 t}}{3} \]

Problem 14851

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+13 y=3 \,{\mathrm e}^{-2 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 t} \cos \left (2 t \right )+\frac {c_{2} {\mathrm e}^{3 t} \sin \left (2 t \right )}{2}+\frac {3 \,{\mathrm e}^{-2 t}}{29} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (\cos \left (2 t \right ) c_{1} +\sin \left (2 t \right ) c_{2} \right ) {\mathrm e}^{-2 t} {\mathrm e}^{5 t}+\frac {3 \,{\mathrm e}^{-2 t}}{29} \]

Problem 14852

ODE	\[ \boxed {y^{\prime \prime }+9 y^{\prime }+20 y=-2 t \,{\mathrm e}^{t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 t}+c_{2} {\mathrm e}^{-4 t}-\frac {t \,{\mathrm e}^{t}}{15}+\frac {11 \,{\mathrm e}^{t}}{450} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {{\mathrm e}^{-5 t} \left (\left (t -\frac {11}{30}\right ) {\mathrm e}^{6 t}-15 \,{\mathrm e}^{t} c_{1} -15 c_{2} \right )}{15} \]

Problem 14853

ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+12 y=3 t^{2} {\mathrm e}^{-4 t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+c_{2} {\mathrm e}^{-3 t}-6 t \,{\mathrm e}^{-4 t}-3 t^{2} {\mathrm e}^{-4 t}-t^{3} {\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-t^{3}-3 t^{2}+c_{1} -6 t \right ) {\mathrm e}^{-4 t}+c_{2} {\mathrm e}^{-3 t} \]

Problem 14854

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-9 y^{\prime }+5 y={\mathrm e}^{t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{t} t +{\mathrm e}^{-5 t} c_{3} +\frac {{\mathrm e}^{t} t^{2}}{12} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\left (12 c_{3} t +t^{2}+12 c_{1} \right ) {\mathrm e}^{6 t}+12 c_{2} \right ) {\mathrm e}^{-5 t}}{12} \]

Problem 14855

ODE	\[ \boxed {y^{\prime \prime \prime }-12 y^{\prime }-16 y={\mathrm e}^{4 t}-{\mathrm e}^{-2 t}} \]
program solution	\[ y = {\mathrm e}^{-2 t} \left (c_{2} t +c_{1} \right )+{\mathrm e}^{4 t} c_{3} +\frac {t^{2} {\mathrm e}^{-2 t}}{12}+\frac {t \,{\mathrm e}^{4 t}}{36} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (18 t^{2}+\left (216 c_{3} +6\right ) t +216 c_{1} +1\right ) {\mathrm e}^{-2 t}}{216}+\frac {\left (t +36 c_{2} -\frac {1}{3}\right ) {\mathrm e}^{4 t}}{36} \]

Problem 14856

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+6 y^{\prime \prime \prime }+18 y^{\prime \prime }+30 y^{\prime }+25 y={\mathrm e}^{-t} \cos \left (2 t \right )+{\mathrm e}^{-2 t} \sin \left (t \right )} \]
program solution	\[ y = {\mathrm e}^{\left (-2+i\right ) t} c_{1} +{\mathrm e}^{\left (-1-2 i\right ) t} c_{2} +{\mathrm e}^{\left (-2-i\right ) t} c_{3} +{\mathrm e}^{\left (-1+2 i\right ) t} c_{4} -\frac {\left (-\frac {1}{20}+2 \left (-\frac {1}{5}+t \right ) \cos \left (\frac {t}{2}\right )^{4}+\sin \left (\frac {t}{2}\right ) \left (t -\frac {11}{5}\right ) \cos \left (\frac {t}{2}\right )^{3}+\left (\left (\frac {1}{5}+t \right ) {\mathrm e}^{-t}-2 t +\frac {2}{5}\right ) \cos \left (\frac {t}{2}\right )^{2}-\frac {\sin \left (\frac {t}{2}\right ) \left (\left (t +\frac {11}{5}\right ) {\mathrm e}^{-t}+t -\frac {11}{5}\right ) \cos \left (\frac {t}{2}\right )}{2}+\frac {\left (-t -\frac {1}{5}\right ) {\mathrm e}^{-t}}{2}+\frac {t}{4}\right ) {\mathrm e}^{-t}}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\left (-20 t +400 c_{3} -6\right ) \cos \left (t \right )^{2}-10 \left (t -40 c_{4} -\frac {21}{5}\right ) \sin \left (t \right ) \cos \left (t \right )+10 t -200 c_{3} +3\right ) {\mathrm e}^{-t}}{200}-\frac {{\mathrm e}^{-2 t} \left (\left (t -10 c_{1} +\frac {7}{10}\right ) \cos \left (t \right )-\frac {\left (t +20 c_{2} +\frac {1}{5}\right ) \sin \left (t \right )}{2}\right )}{10} \]

Problem 14857

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+14 y^{\prime \prime }+20 y^{\prime }+25 y=t^{2}} \]
program solution	\[ y = \left (c_{2} t +c_{1} \right ) {\mathrm e}^{\left (-1-2 i\right ) t}+{\mathrm e}^{\left (-1+2 i\right ) t} \left (c_{4} t +c_{3} \right )+\frac {t^{2}}{25}-\frac {8 t}{125}+\frac {4}{625} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {4}{625}+\left (\left (c_{3} t +c_{1} \right ) \cos \left (2 t \right )+\sin \left (2 t \right ) \left (t c_{4} +c_{2} \right )\right ) {\mathrm e}^{-t}+\frac {t^{2}}{25}-\frac {8 t}{125} \]

Problem 14858

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

Problem 14859

ODE	\[ \boxed {y^{\prime \prime }+10 y^{\prime }+16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = \frac {2 \,{\mathrm e}^{-2 t}}{3}-\frac {2 \,{\mathrm e}^{-8 t}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {2 \,{\mathrm e}^{-8 t}}{3}+\frac {2 \,{\mathrm e}^{-2 t}}{3} \]

Problem 14860

ODE	\[ \boxed {y^{\prime \prime }+16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -8] \end {align*}
program solution	\[ y = -2 \sin \left (4 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -2 \sin \left (4 t \right ) \]

Problem 14861

ODE	\[ \boxed {y^{\prime \prime }+25 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \cos \left (5 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \cos \left (5 t \right ) \]

Problem 14862

ODE	\[ \boxed {y^{\prime \prime }-4 y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {17 \,{\mathrm e}^{2 t}}{16}+\frac {15 \,{\mathrm e}^{-2 t}}{16}-\frac {t}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {17 \,{\mathrm e}^{2 t}}{16}+\frac {15 \,{\mathrm e}^{-2 t}}{16}-\frac {t}{4} \]

Problem 14863

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-4 y={\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = \frac {19 \,{\mathrm e}^{t}}{25}-\frac {19 \,{\mathrm e}^{-4 t}}{25}+\frac {t \,{\mathrm e}^{t}}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (5 t +19\right ) {\mathrm e}^{-4 t} {\mathrm e}^{5 t}}{25}-\frac {19 \,{\mathrm e}^{-4 t}}{25} \]

Problem 14864

ODE	\[ \boxed {y^{\prime \prime }+9 y=\sin \left (3 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 6 \cos \left (3 t \right )+\frac {\sin \left (3 t \right )}{18}-\frac {\cos \left (3 t \right ) t}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sin \left (3 t \right )}{18}+6 \cos \left (3 t \right )-\frac {\cos \left (3 t \right ) t}{6} \]

Problem 14865

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

Problem 14866

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

Problem 14867

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

Problem 14868

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+16 y=\frac {{\mathrm e}^{4 t}}{t^{3}}} \]
program solution	\[ y = {\mathrm e}^{4 t} \left (c_{2} t +c_{1} \right )+\frac {{\mathrm e}^{4 t}}{2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{4 t} \left (2 c_{1} t^{2}+2 t c_{2} +1\right )}{2 t} \]

Problem 14869

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+16 y=\frac {{\mathrm e}^{4 t}}{t^{3}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 14870

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y={\mathrm e}^{t} \ln \left (t \right )} \]
program solution	\[ y = {\mathrm e}^{t} \left (c_{2} t +c_{1} \right )+\frac {{\mathrm e}^{t} t^{2} \left (-3+2 \ln \left (t \right )\right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{t} \left (2 \ln \left (t \right ) t^{2}+4 c_{1} t -3 t^{2}+4 c_{2} \right )}{4} \]

Problem 14871

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y={\mathrm e}^{t} \ln \left (t \right )} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = \left (2-t \right ) {\mathrm e}^{t -1}+\frac {{\mathrm e}^{t} \left (t^{2} \ln \left (t \right )-\frac {3 t^{2}}{2}+2 t -\frac {1}{2}\right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{t} \left (2 \ln \left (t \right ) t^{2}-4 \,{\mathrm e}^{-1} t -3 t^{2}+8 \,{\mathrm e}^{-1}+4 t -1\right )}{4} \]

Problem 14872

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

Problem 14873

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }-4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 t}+\frac {c_{2} {\mathrm e}^{t}}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{1} {\mathrm e}^{5 t}+c_{2} \right ) {\mathrm e}^{-4 t} \]

Problem 14874

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{-2 t} t \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-2 t} \left (t c_{2} +c_{1} \right ) \]

Problem 14875

ODE	\[ \boxed {t^{2} y^{\prime \prime }-5 t y^{\prime }+5 y=0} \]
program solution	\[ y = t c_{1} +\frac {1}{4} c_{2} t^{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = t \left (c_{1} t^{4}+c_{2} \right ) \]

Problem 14876

ODE	\[ \boxed {x^{2} y^{\prime \prime }+7 y^{\prime } x +8 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{4}}+\frac {c_{2}}{2 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2}+c_{2}}{x^{4}} \]

Problem 14877

ODE	\[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +6 y=0} \]
program solution	\[ y = c_{2} x^{3}+c_{1} x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{1} x +c_{2} \right ) \]

Problem 14878

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y=0} \]
program solution	\[ y = x^{-i} c_{1} -\frac {i c_{2} x^{i}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\ln \left (x \right )\right )+c_{2} \cos \left (\ln \left (x \right )\right ) \]

Problem 14879

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+5 y^{\prime } x +y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {2 x -\sqrt {x}}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2}{\sqrt {x}}-\frac {1}{x} \]

Problem 14880

ODE	\[ \boxed {5 x^{2} y^{\prime \prime }-y^{\prime } x +2 y=0} \]
program solution	\[ y = c_{1} x^{\frac {3}{5}-\frac {i}{5}}-\frac {5 i c_{2} x^{\frac {3}{5}+\frac {i}{5}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{\frac {3}{5}} \left (c_{1} \sin \left (\frac {\ln \left (x \right )}{5}\right )+c_{2} \cos \left (\frac {\ln \left (x \right )}{5}\right )\right ) \]

Problem 14881

ODE	\[ \boxed {x^{2} y^{\prime \prime }-7 y^{\prime } x +25 y=0} \]
program solution	\[ y = c_{1} x^{4-3 i}-\frac {i c_{2} x^{4+3 i}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{4} \left (c_{1} \sin \left (3 \ln \left (x \right )\right )+c_{2} \cos \left (3 \ln \left (x \right )\right )\right ) \]

Problem 14882

ODE	\[ \boxed {x^{2} y^{\prime \prime }-7 y^{\prime } x +15 y=8 x} \]
program solution	\[ y = c_{1} x^{3}+\frac {1}{2} c_{2} x^{5}+x \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} x^{5}+c_{1} x^{3}+x \]

Problem 14883

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-2 x^{2}-\frac {8}{3} x^{3}-2 x^{4}-\frac {16}{15} x^{5}-\frac {4}{9} x^{6}\right ) y \left (0\right )+\left (x +2 x^{2}+2 x^{3}+\frac {4}{3} x^{4}+\frac {2}{3} x^{5}+\frac {4}{15} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-2 x^{2}-\frac {8}{3} x^{3}-2 x^{4}-\frac {16}{15} x^{5}\right ) c_{1} +\left (x +2 x^{2}+2 x^{3}+\frac {4}{3} x^{4}+\frac {2}{3} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-2 x^{2}-\frac {8}{3} x^{3}-2 x^{4}-\frac {16}{15} x^{5}\right ) y \left (0\right )+\left (x +2 x^{2}+2 x^{3}+\frac {4}{3} x^{4}+\frac {2}{3} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14884

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-3 y=x \,{\mathrm e}^{x}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {3}{2} x^{2}-x^{3}+\frac {7}{8} x^{4}-\frac {1}{2} x^{5}+\frac {61}{240} x^{6}\right ) y \left (0\right )+\left (x -x^{2}+\frac {7}{6} x^{3}-\frac {5}{6} x^{4}+\frac {61}{120} x^{5}-\frac {91}{360} x^{6}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}+\frac {x^{5}}{20}-\frac {x^{6}}{90}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {3}{2} x^{2}-x^{3}+\frac {7}{8} x^{4}-\frac {1}{2} x^{5}\right ) c_{1} +\left (x -x^{2}+\frac {7}{6} x^{3}-\frac {5}{6} x^{4}+\frac {61}{120} x^{5}\right ) c_{2} +\frac {x^{3}}{6}+\frac {x^{5}}{20}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3}{2} x^{2}-x^{3}+\frac {7}{8} x^{4}-\frac {1}{2} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {7}{6} x^{3}-\frac {5}{6} x^{4}+\frac {61}{120} x^{5}\right ) D\left (y \right )\left (0\right )+\frac {x^{3}}{6}+\frac {x^{5}}{20}+O\left (x^{6}\right ) \]

Problem 14885

ODE	\[ \boxed {\left (2 x^{2}-1\right ) y^{\prime \prime }+2 y^{\prime } x -3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{2} x^{2}-\frac {5}{8} x^{4}-\frac {29}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{8} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{2} x^{2}-\frac {5}{8} x^{4}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}-\frac {1}{8} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{2} x^{2}-\frac {5}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{8} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 14886

ODE	\[ \boxed {3 x y^{\prime \prime }+11 y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x}{11}+\frac {x^{2}}{308}+\frac {x^{3}}{15708}+\frac {x^{4}}{1256640}+\frac {x^{5}}{144513600}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x}{5}+\frac {x^{2}}{20}+\frac {x^{3}}{60}+\frac {x^{4}}{960}+\frac {x^{5}}{33600}+O\left (x^{6}\right )\right )}{x^{\frac {8}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{5} x +\frac {1}{20} x^{2}+\frac {1}{60} x^{3}+\frac {1}{960} x^{4}+\frac {1}{33600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {8}{3}}}+c_{2} \left (1+\frac {1}{11} x +\frac {1}{308} x^{2}+\frac {1}{15708} x^{3}+\frac {1}{1256640} x^{4}+\frac {1}{144513600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 14887

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

Problem 14888

ODE	\[ \boxed {x^{2} y^{\prime \prime }-7 y^{\prime } x +\left (-2 x^{2}+7\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{7} \left (1+\frac {x^{2}}{8}+\frac {x^{4}}{160}+\frac {x^{6}}{5760}+O\left (x^{7}\right )\right )+c_{2} \left (\frac {x^{7} \left (1+\frac {x^{2}}{8}+\frac {x^{4}}{160}+\frac {x^{6}}{5760}+O\left (x^{7}\right )\right ) \ln \left (x \right )}{48}+x \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{16}+O\left (x^{7}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{7} \left (1+\frac {1}{8} x^{2}+\frac {1}{160} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (-86400+21600 x^{2}-5400 x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 14889

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (2 x +1\right ) y^{\prime }+10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (-6 x^{5}+25 x^{4}-40 x^{3}+30 x^{2}-10 x +1+O\left (x^{6}\right )\right )+c_{2} \left (\left (-6 x^{5}+25 x^{4}-40 x^{3}+30 x^{2}-10 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+17 x -\frac {157 x^{2}}{2}+\frac {404 x^{3}}{3}-\frac {625 x^{4}}{6}+\frac {162 x^{5}}{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-10 x +30 x^{2}-40 x^{3}+25 x^{4}-6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (17 x -\frac {157}{2} x^{2}+\frac {404}{3} x^{3}-\frac {625}{6} x^{4}+\frac {162}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 14890

ODE	\[ \boxed {x \left (x +1\right ) y^{\prime \prime }+\left (1-2 x \right ) y^{\prime }-10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (6 x^{5}+25 x^{4}+40 x^{3}+30 x^{2}+10 x +1+O\left (x^{6}\right )\right )+c_{2} \left (\left (6 x^{5}+25 x^{4}+40 x^{3}+30 x^{2}+10 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )-17 x -\frac {157 x^{2}}{2}-\frac {404 x^{3}}{3}-\frac {625 x^{4}}{6}-\frac {162 x^{5}}{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+10 x +30 x^{2}+40 x^{3}+25 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-17\right ) x -\frac {157}{2} x^{2}-\frac {404}{3} x^{3}-\frac {625}{6} x^{4}-\frac {162}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 14891

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

Problem 14892

ODE	\[ \boxed {4 x^{\prime \prime }+9 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = -1, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = -\cos \left (\frac {3 t}{2}\right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\cos \left (\frac {3 t}{2}\right ) \]

Problem 14893

ODE	\[ \boxed {9 x^{\prime \prime }+4 x=0} \] With initial conditions \begin {align*} \left [x \left (0\right ) = -{\frac {1}{2}}, x^{\prime }\left (0\right ) = 1\right ] \end {align*}
program solution	\[ x = -\frac {\cos \left (\frac {2 t}{3}\right )}{2}+\frac {3 \sin \left (\frac {2 t}{3}\right )}{2} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {3 \sin \left (\frac {2 t}{3}\right )}{2}-\frac {\cos \left (\frac {2 t}{3}\right )}{2} \]

Problem 14894

ODE	\[ \boxed {x^{\prime \prime }+64 x=0} \] With initial conditions \begin {align*} \left [x \left (0\right ) = {\frac {3}{4}}, x^{\prime }\left (0\right ) = -2\right ] \end {align*}
program solution	\[ x = \frac {3 \cos \left (8 t \right )}{4}-\frac {\sin \left (8 t \right )}{4} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\frac {\sin \left (8 t \right )}{4}+\frac {3 \cos \left (8 t \right )}{4} \]

Problem 14895

ODE	\[ \boxed {x^{\prime \prime }+100 x=0} \] With initial conditions \begin {align*} \left [x \left (0\right ) = -{\frac {1}{4}}, x^{\prime }\left (0\right ) = 1\right ] \end {align*}
program solution	\[ x = -\frac {\cos \left (10 t \right )}{4}+\frac {\sin \left (10 t \right )}{10} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\sin \left (10 t \right )}{10}-\frac {\cos \left (10 t \right )}{4} \]

Problem 14896

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

Problem 14897

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

Problem 14898

ODE	\[ \boxed {x^{\prime \prime }+16 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = -2, x^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ x = \frac {\sin \left (4 t \right )}{4}-2 \cos \left (4 t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\sin \left (4 t \right )}{4}-2 \cos \left (4 t \right ) \]

Problem 14899

ODE	\[ \boxed {x^{\prime \prime }+256 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 2, x^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ x = \frac {\sin \left (16 t \right )}{4}+2 \cos \left (16 t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\sin \left (16 t \right )}{4}+2 \cos \left (16 t \right ) \]

Problem 14900

ODE	\[ \boxed {x^{\prime \prime }+9 x=0} \] With initial conditions \begin {align*} \left [x \left (0\right ) = {\frac {1}{3}}, x^{\prime }\left (0\right ) = -1\right ] \end {align*}
program solution	\[ x = \frac {\cos \left (3 t \right )}{3}-\frac {\sin \left (3 t \right )}{3} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\frac {\sin \left (3 t \right )}{3}+\frac {\cos \left (3 t \right )}{3} \]