2.17.25 Problems 2401 to 2500

Problem 2401

ODE	\[ \boxed {2 x y^{\prime \prime }-\left (x^{3}+1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x}{5}+\frac {x^{2}}{70}+\frac {52 x^{3}}{945}-\frac {1049 x^{4}}{83160}+\frac {5207 x^{5}}{5405400}+O\left (x^{6}\right )\right )+c_{2} \left (1+x -\frac {x^{2}}{2}+\frac {x^{3}}{18}+\frac {17 x^{4}}{360}-\frac {377 x^{5}}{12600}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {3}{2}} \left (1-\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {52}{945} x^{3}-\frac {1049}{83160} x^{4}+\frac {5207}{5405400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+x -\frac {1}{2} x^{2}+\frac {1}{18} x^{3}+\frac {17}{360} x^{4}-\frac {377}{12600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2402

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	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +x^{2}-\frac {2}{9} x^{3}+\frac {1}{36} x^{4}-\frac {1}{450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (4 x -3 x^{2}+\frac {22}{27} x^{3}-\frac {25}{216} x^{4}+\frac {137}{13500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2403

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{16}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {x^{2}}{2}+\frac {x^{4}}{16}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{2}-\frac {3 x^{4}}{32}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{2} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x^{2}-\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2404

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +4 y \left (x +1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (4 x^{2}-4 x +1-\frac {16 x^{3}}{9}+\frac {4 x^{4}}{9}-\frac {16 x^{5}}{225}+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (4 x^{2}-4 x +1-\frac {16 x^{3}}{9}+\frac {4 x^{4}}{9}-\frac {16 x^{5}}{225}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-12 x^{2}+8 x +\frac {176 x^{3}}{27}-\frac {50 x^{4}}{27}+\frac {1096 x^{5}}{3375}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-4 x +4 x^{2}-\frac {16}{9} x^{3}+\frac {4}{9} x^{4}-\frac {16}{225} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (8 x -12 x^{2}+\frac {176}{27} x^{3}-\frac {50}{27} x^{4}+\frac {1096}{3375} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{2} \]

Problem 2405

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (x +1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (x +1+\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (x +1+\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (-x -\frac {3 x^{2}}{4}-\frac {11 x^{3}}{36}-\frac {25 x^{4}}{288}-\frac {137 x^{5}}{7200}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x -\frac {3}{4} x^{2}-\frac {11}{36} x^{3}-\frac {25}{288} x^{4}-\frac {137}{7200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x \]

Problem 2406

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (2 x +3\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (6 x^{2}+4 x +1+\frac {16 x^{3}}{3}+\frac {10 x^{4}}{3}+\frac {8 x^{5}}{5}+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (6 x^{2}+4 x +1+\frac {16 x^{3}}{3}+\frac {10 x^{4}}{3}+\frac {8 x^{5}}{5}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-13 x^{2}-6 x -\frac {124 x^{3}}{9}-\frac {173 x^{4}}{18}-\frac {374 x^{5}}{75}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+4 x +6 x^{2}+\frac {16}{3} x^{3}+\frac {10}{3} x^{4}+\frac {8}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-6\right ) x -13 x^{2}-\frac {124}{9} x^{3}-\frac {173}{18} x^{4}-\frac {374}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{2} \]

Problem 2407

ODE	\[ \boxed {x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }-5 y^{\prime } x +9 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+\frac {3 x^{2}}{2}+\frac {15 x^{4}}{8}+O\left (x^{6}\right )\right )+c_{2} \left (x^{3} \left (1+\frac {3 x^{2}}{2}+\frac {15 x^{4}}{8}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{3} \left (-\frac {x^{2}}{4}-\frac {13 x^{4}}{32}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {3}{2} x^{2}+\frac {15}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {13}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{3} \]

Problem 2408

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

Problem 2409

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (2 x -1\right ) y^{\prime }+x \left (x -1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-x +\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{24}-\frac {x^{5}}{120}+O\left (x^{6}\right )\right )+c_{2} \left (1-x +\frac {x^{3}}{3}-\frac {5 x^{4}}{24}+\frac {3 x^{5}}{40}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-2+2 x -\frac {2}{3} x^{3}+\frac {5}{12} x^{4}-\frac {3}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2410

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x^{2}+\left (x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{60}-\frac {x^{4}}{210}-\frac {x^{5}}{3360}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{2}+\frac {5 x^{3}}{12}+\frac {x^{4}}{12}-\frac {x^{5}}{60}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{60} x^{3}-\frac {1}{210} x^{4}-\frac {1}{3360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+6 x +6 x^{2}+5 x^{3}+x^{4}-\frac {1}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2411

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x^{2}-\left (3 x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-x +\frac {9 x^{2}}{10}-\frac {17 x^{3}}{30}+\frac {251 x^{4}}{840}-\frac {37 x^{5}}{280}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {3 x^{2}}{2}+\frac {11 x^{3}}{3}-\frac {115 x^{4}}{24}+\frac {159 x^{5}}{40}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-x +\frac {9}{10} x^{2}-\frac {17}{30} x^{3}+\frac {251}{840} x^{4}-\frac {37}{280} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-12 x -18 x^{2}+44 x^{3}-\frac {115}{2} x^{4}+\frac {477}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2412

ODE	\[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +1\right ) y^{\prime }-9 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+\frac {3 x}{7}+\frac {3 x^{2}}{14}+\frac {5 x^{3}}{42}+\frac {x^{4}}{14}+\frac {x^{5}}{22}+\frac {x^{6}}{33}+O\left (x^{7}\right )\right )+\frac {c_{2} \left (1-3 x +3 x^{2}-x^{3}+O\left (x^{7}\right )\right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+\frac {3}{7} x +\frac {3}{14} x^{2}+\frac {5}{42} x^{3}+\frac {1}{14} x^{4}+\frac {1}{22} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400+259200 x -259200 x^{2}+86400 x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

Problem 2413

ODE	\[ \boxed {\left (-x^{2}+x \right ) y^{\prime \prime }-3 y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{3}-\frac {x^{4}}{3}-\frac {2 x^{5}}{3}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-144-96 x -48 x^{2}+48 x^{4}+96 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2414

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x -7\right ) y^{\prime }+\left (x +12\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{6} \left (1-\frac {7 x}{5}+\frac {14 x^{2}}{15}-\frac {2 x^{3}}{5}+\frac {x^{4}}{8}-\frac {11 x^{5}}{360}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {5 x^{6} \left (1-\frac {7 x}{5}+\frac {14 x^{2}}{15}-\frac {2 x^{3}}{5}+\frac {x^{4}}{8}-\frac {11 x^{5}}{360}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+x^{2} \left (1+x +x^{2}+\frac {5 x^{3}}{3}-\frac {37 x^{5}}{10}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x^{4} \left (1-\frac {7}{5} x +\frac {14}{15} x^{2}-\frac {2}{5} x^{3}+\frac {1}{8} x^{4}-\frac {11}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (360 x^{4}-504 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-144 x -144 x^{2}-240 x^{3}+342 x^{4}+54 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x^{2} \]

Problem 2415

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (x -4\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+O\left (x^{6}\right )\right )+c_{2} \left (-3 x^{4} \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x \left (1+\frac {x}{2}+x^{2}-9 x^{4}+\frac {63 x^{5}}{2}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x^{3} \left (1-4 x +10 x^{2}-20 x^{3}+35 x^{4}-56 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-36\right ) x^{3}+144 x^{4}-360 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+6 x +12 x^{2}-240 x^{3}+852 x^{4}-2022 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x \]

Problem 2416

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (-x^{2}+3\right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x^{2}}{12}+\frac {x^{4}}{128}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {3 x \left (1+\frac {x^{2}}{12}+\frac {x^{4}}{128}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{16}+\frac {1+\frac {3 x^{2}}{4}+O\left (x^{6}\right )}{x^{3}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1+\frac {1}{12} x^{2}+\frac {1}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (27 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-108 x^{2}-36 x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{3}} \]

Problem 2417

ODE	\[ \boxed {x y^{\prime \prime }+3 y^{\prime }-y=x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{8}+\frac {x^{3}}{120}+\frac {x^{4}}{2880}+\frac {x^{5}}{100800}+O\left (x^{6}\right )+c_{1} \left (1+\frac {x}{3}+\frac {x^{2}}{24}+\frac {x^{3}}{360}+\frac {x^{4}}{8640}+\frac {x^{5}}{302400}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-\frac {1}{2}-\frac {x}{6}-\frac {x^{2}}{48}-\frac {x^{3}}{720}-\frac {x^{4}}{17280}-\frac {x^{5}}{604800}-\frac {O\left (x^{6}\right )}{2}\right ) \ln \left (x \right )+\frac {1-x +\frac {2 x^{3}}{9}+\frac {25 x^{4}}{576}+\frac {157 x^{5}}{43200}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {1}{3} x +\frac {1}{24} x^{2}+\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}+\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (x^{2}+\frac {1}{3} x^{3}+\frac {1}{24} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {4}{9} x^{3}-\frac {25}{288} x^{4}-\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}}+x^{2} \left (\frac {1}{8}+\frac {1}{120} x +\frac {1}{2880} x^{2}+\frac {1}{100800} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 2418

ODE	\[ \boxed {x y^{\prime \prime }+3 y^{\prime }-y=x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{8}+\frac {x^{3}}{120}+\frac {x^{4}}{2880}+\frac {x^{5}}{100800}+O\left (x^{6}\right )+c_{1} \left (1+\frac {x}{3}+\frac {x^{2}}{24}+\frac {x^{3}}{360}+\frac {x^{4}}{8640}+\frac {x^{5}}{302400}+O\left (x^{6}\right )\right )+c_{2} \left (\left (-\frac {1}{2}-\frac {x}{6}-\frac {x^{2}}{48}-\frac {x^{3}}{720}-\frac {x^{4}}{17280}-\frac {x^{5}}{604800}-\frac {O\left (x^{6}\right )}{2}\right ) \ln \left (x \right )+\frac {1-x +\frac {2 x^{3}}{9}+\frac {25 x^{4}}{576}+\frac {157 x^{5}}{43200}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {1}{3} x +\frac {1}{24} x^{2}+\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}+\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (x^{2}+\frac {1}{3} x^{3}+\frac {1}{24} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+2 x -\frac {4}{9} x^{3}-\frac {25}{288} x^{4}-\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}}+x^{2} \left (\frac {1}{8}+\frac {1}{120} x +\frac {1}{2880} x^{2}+\frac {1}{100800} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 2419

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }-2 y x=x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{3}}{9}+\frac {2 x^{5}}{225}+O\left (x^{6}\right )+c_{1} \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{16}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1+\frac {x^{2}}{2}+\frac {x^{4}}{16}+O\left (x^{6}\right )\right ) \ln \left (x \right )-\frac {x^{2}}{2}-\frac {3 x^{4}}{32}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x^{2}+\frac {1}{16} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{3} \left (\frac {1}{9}+\frac {2}{225} x^{2}+\operatorname {O}\left (x^{3}\right )\right )+\left (-\frac {1}{2} x^{2}-\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2420

ODE	\[ \boxed {x y^{\prime \prime }-y^{\prime } x +y=x^{3}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{4}}{12}+\frac {x^{5}}{80}+O\left (x^{6}\right )+c_{1} x \left (1+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {x^{2}}{2}-\frac {x^{3}}{12}-\frac {x^{4}}{72}-\frac {x^{5}}{480}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{4} \left (\frac {1}{12}+\frac {1}{80} x +\operatorname {O}\left (x^{2}\right )\right )+\ln \left (x \right ) \left (-x +\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1+x -\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{72} x^{4}-\frac {1}{480} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 2421

ODE	\[ \boxed {\left (-2 x +1\right ) y^{\prime \prime }+4 y^{\prime } x -4 y=x^{2}-x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}+\frac {4}{45} x^{6}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x -\frac {x^{3}}{6}-\frac {x^{4}}{12}-\frac {x^{5}}{30}-\frac {x^{6}}{90}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}\right ) c_{1} +c_{2} x -\frac {x^{3}}{6}-\frac {x^{4}}{12}-\frac {x^{5}}{30}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}\right ) y \left (0\right )+D\left (y \right )\left (0\right ) x -\frac {x^{3}}{6}-\frac {x^{4}}{12}-\frac {x^{5}}{30}+O\left (x^{6}\right ) \]

Problem 2422

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x +12\right ) y=x^{2}+x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x}{13}+\frac {3 x^{2}}{52}-\frac {x^{3}}{364}+\frac {x^{4}}{10192}-\frac {x^{5}}{377104}+O\left (x^{6}\right )+c_{1} x^{2 i \sqrt {3}} \left (1+\frac {x}{-1-4 i \sqrt {3}}-\frac {x^{2}}{4 \left (i-4 \sqrt {3}\right ) \left (-2 \sqrt {3}+i\right )}+\frac {x^{3}}{48 \left (-2 \sqrt {3}+i\right ) \left (i \sqrt {3}+\frac {3}{4}\right ) \left (i-4 \sqrt {3}\right )}+\frac {x^{4}}{192 \left (4 \sqrt {3}-3 i\right ) \left (-i+\sqrt {3}\right ) \left (2 \sqrt {3}-i\right ) \left (-i+4 \sqrt {3}\right )}+\frac {x^{5}}{15360 \left (\sqrt {3}-\frac {3 i}{4}\right ) \left (-i+\sqrt {3}\right ) \left (-2 \sqrt {3}+i\right ) \left (-i+4 \sqrt {3}\right ) \left (i \sqrt {3}+\frac {5}{4}\right )}+O\left (x^{6}\right )\right )+c_{2} x^{-2 i \sqrt {3}} \left (1+\frac {x}{-1+4 i \sqrt {3}}-\frac {x^{2}}{4 \left (-i-4 \sqrt {3}\right ) \left (-2 \sqrt {3}-i\right )}+\frac {x^{3}}{48 \left (-2 \sqrt {3}-i\right ) \left (-i \sqrt {3}+\frac {3}{4}\right ) \left (-i-4 \sqrt {3}\right )}+\frac {x^{4}}{192 \left (4 \sqrt {3}+3 i\right ) \left (\sqrt {3}+i\right ) \left (2 \sqrt {3}+i\right ) \left (i+4 \sqrt {3}\right )}+\frac {x^{5}}{15360 \left (\sqrt {3}+\frac {3 i}{4}\right ) \left (\sqrt {3}+i\right ) \left (-2 \sqrt {3}-i\right ) \left (i+4 \sqrt {3}\right ) \left (-i \sqrt {3}+\frac {5}{4}\right )}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} x^{2 i \sqrt {3}} \left (1+\frac {1}{-4 i \sqrt {3}-1} x -\frac {1}{4} \frac {1}{\left (i-2 \sqrt {3}\right ) \left (-4 \sqrt {3}+i\right )} x^{2}+\frac {1}{48} \frac {1}{\left (i-2 \sqrt {3}\right ) \left (-4 \sqrt {3}+i\right ) \left (i \sqrt {3}+\frac {3}{4}\right )} x^{3}+\frac {1}{768} \frac {1}{\left (-\sqrt {3}+\frac {3 i}{4}\right ) \left (-i+2 \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \left (-4 \sqrt {3}+i\right )} x^{4}+\frac {1}{15360} \frac {1}{\left (\sqrt {3}-\frac {3 i}{4}\right ) \left (-i+2 \sqrt {3}\right ) \left (-i+\sqrt {3}\right ) \left (-4 \sqrt {3}+i\right ) \left (i \sqrt {3}+\frac {5}{4}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-2 i \sqrt {3}} \left (1+\frac {1}{4 i \sqrt {3}-1} x -\frac {1}{4} \frac {1}{\left (2 \sqrt {3}+i\right ) \left (4 \sqrt {3}+i\right )} x^{2}-\frac {1}{48} \frac {1}{\left (2 \sqrt {3}+i\right ) \left (4 \sqrt {3}+i\right ) \left (i \sqrt {3}-\frac {3}{4}\right )} x^{3}+\frac {1}{192} \frac {1}{\left (2 \sqrt {3}+i\right ) \left (3 i+4 \sqrt {3}\right ) \left (\sqrt {3}+i\right ) \left (4 \sqrt {3}+i\right )} x^{4}+\frac {1}{960} \frac {1}{\left (4 i \sqrt {3}-3\right ) \left (2 \sqrt {3}+i\right ) \left (\sqrt {3}+i\right ) \left (5 i+4 \sqrt {3}\right ) \left (4 \sqrt {3}+i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x \left (\frac {1}{13}+\frac {3}{52} x -\frac {1}{364} x^{2}+\frac {1}{10192} x^{3}-\frac {1}{377104} x^{4}+\operatorname {O}\left (x^{5}\right )\right ) \]

Problem 2423

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (x^{2}+3\right ) y^{\prime }+y=-2 x^{2}+x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x}{4}-\frac {2 x^{2}}{9}+\frac {7 x^{3}}{576}+\frac {107 x^{4}}{7200}-\frac {1031 x^{5}}{172800}+O\left (x^{6}\right )+\frac {c_{1} \left (-2 x +1+\frac {x^{2}}{4}-\frac {x^{4}}{64}+\frac {3 x^{5}}{800}+O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {\left (-2 x +1+\frac {x^{2}}{4}-\frac {x^{4}}{64}+\frac {3 x^{5}}{800}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {-x^{2}+7 x +\frac {7 x^{3}}{36}+\frac {35 x^{4}}{1152}-\frac {191 x^{5}}{9000}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2} \left (\frac {1}{4}-\frac {2}{9} x +\frac {7}{576} x^{2}+\frac {107}{7200} x^{3}-\frac {1031}{172800} x^{4}+\operatorname {O}\left (x^{5}\right )\right )+\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +\frac {1}{4} x^{2}-\frac {1}{64} x^{4}+\frac {3}{800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (7 x -x^{2}+\frac {7}{36} x^{3}+\frac {35}{1152} x^{4}-\frac {191}{9000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x} \]

Problem 2424

ODE	\[ \boxed {3 x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (5-x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y=-x^{3}+x} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x}{4}+\frac {x^{2}}{60}-\frac {47 x^{3}}{960}+\frac {673 x^{4}}{52800}-\frac {1169 x^{5}}{316800}+O\left (x^{6}\right )+c_{1} x^{\frac {1}{3}} \left (1+\frac {x}{7}-\frac {x^{2}}{10}+\frac {29 x^{3}}{2730}-\frac {17 x^{4}}{87360}-\frac {1193 x^{5}}{8299200}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+7 x -\frac {x^{2}}{2}-\frac {29 x^{3}}{30}+\frac {73 x^{4}}{480}-\frac {167 x^{5}}{26400}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1+\frac {1}{7} x -\frac {1}{10} x^{2}+\frac {29}{2730} x^{3}-\frac {17}{87360} x^{4}-\frac {1193}{8299200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{4}+\frac {1}{60} x -\frac {47}{960} x^{2}+\frac {673}{52800} x^{3}-\frac {1169}{316800} x^{4}+\operatorname {O}\left (x^{5}\right )\right )+c_{1} \left (1+7 x -\frac {1}{2} x^{2}-\frac {29}{30} x^{3}+\frac {73}{480} x^{4}-\frac {167}{26400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2425

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+\left (2+3 x \right ) y=x^{4}+x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{20}-\frac {3 x^{3}}{1120}+\frac {1129 x^{4}}{123200}-\frac {3387 x^{5}}{22422400}+O\left (x^{6}\right )+c_{1} x^{\frac {2}{3}} \left (1-\frac {x}{4}+\frac {x^{2}}{56}-\frac {x^{3}}{1680}+\frac {x^{4}}{87360}-\frac {x^{5}}{6988800}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{480}+\frac {x^{4}}{21120}-\frac {x^{5}}{1478400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{480} x^{3}+\frac {1}{21120} x^{4}-\frac {1}{1478400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {2}{3}} \left (1-\frac {1}{4} x +\frac {1}{56} x^{2}-\frac {1}{1680} x^{3}+\frac {1}{87360} x^{4}-\frac {1}{6988800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{20}-\frac {3}{1120} x +\frac {1129}{123200} x^{2}-\frac {3387}{22422400} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 2426

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+10 y^{\prime } x +y=x -1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -1+\frac {x}{11}+O\left (x^{6}\right )+c_{1} x^{-\frac {1}{18}+\frac {i \sqrt {35}}{18}} \left (1+O\left (x^{6}\right )\right )+c_{2} x^{-\frac {1}{18}-\frac {i \sqrt {35}}{18}} \left (1+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-\frac {1}{18}-\frac {i \sqrt {35}}{18}} \left (1+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{-\frac {1}{18}+\frac {i \sqrt {35}}{18}} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (-1+\frac {1}{11} x +\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2427

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+\left (-x^{2}+x \right ) y^{\prime }-y=x^{3}+1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -1-\frac {x}{3}-\frac {x^{2}}{15}+\frac {13 x^{3}}{210}+\frac {13 x^{4}}{1890}+\frac {13 x^{5}}{20790}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x}{5}+\frac {x^{2}}{35}+\frac {x^{3}}{315}+\frac {x^{4}}{3465}+\frac {x^{5}}{45045}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{8}+\frac {x^{3}}{48}+\frac {x^{4}}{384}+\frac {x^{5}}{3840}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x +\frac {1}{35} x^{2}+\frac {1}{315} x^{3}+\frac {1}{3465} x^{4}+\frac {1}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-1+\frac {1}{14} x^{3}+\frac {1}{126} x^{4}+\frac {1}{1386} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2428

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

Problem 2429

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

Problem 2430

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+5 y^{\prime } x +y \left (x +1\right )=x \left (x^{2}+x +1\right )} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x}{6}+\frac {x^{2}}{18}+\frac {17 x^{3}}{504}-\frac {17 x^{4}}{22680}+\frac {17 x^{5}}{1496880}+O\left (x^{6}\right )+\frac {c_{1} \left (1-\frac {x}{3}+\frac {x^{2}}{30}-\frac {x^{3}}{630}+\frac {x^{4}}{22680}-\frac {x^{5}}{1247400}+O\left (x^{6}\right )\right )}{\sqrt {x}}+\frac {c_{2} \left (1-x +\frac {x^{2}}{6}-\frac {x^{3}}{90}+\frac {x^{4}}{2520}-\frac {x^{5}}{113400}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+x \left (\frac {1}{6}+\frac {1}{18} x +\frac {17}{504} x^{2}-\frac {17}{22680} x^{3}+\frac {17}{1496880} x^{4}+\operatorname {O}\left (x^{5}\right )\right ) \]

Problem 2431

ODE	\[ \boxed {\left (x^{3}+2 x^{2}\right ) y^{\prime \prime }-y^{\prime } x +\left (1-x \right ) y=x^{2} \left (x +1\right )^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{3}+\frac {x^{3}}{6}+\frac {x^{4}}{126}-\frac {11 x^{5}}{4536}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x}{3}-\frac {x^{2}}{30}+\frac {x^{3}}{126}-\frac {11 x^{4}}{4536}+\frac {19 x^{5}}{22680}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {5 x}{4}+\frac {5 x^{2}}{96}-\frac {11 x^{3}}{1152}+\frac {341 x^{4}}{129024}-\frac {20119 x^{5}}{23224320}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {5}{4} x +\frac {5}{96} x^{2}-\frac {11}{1152} x^{3}+\frac {341}{129024} x^{4}-\frac {20119}{23224320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{3} x -\frac {1}{30} x^{2}+\frac {1}{126} x^{3}-\frac {11}{4536} x^{4}+\frac {19}{22680} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{6} x +\frac {1}{126} x^{2}-\frac {11}{4536} x^{3}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 2432

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

Problem 2433

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

Problem 2434

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

Problem 2435

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

Problem 2436

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

Problem 2437

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

Problem 2438

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

Problem 2439

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

Problem 2440

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

Problem 2441

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

Problem 2442

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

Problem 2443

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

Problem 2444

ODE	\[ \boxed {{y^{\prime }}^{2}-3 y^{\prime }=-2} \]
program solution	\[ y = 2 x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 2 x +c_{1} \\ y \left (x \right ) &= c_{1} +x \\ \end{align*}

Problem 2445

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

Problem 2446

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

Problem 2447

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

Problem 2448

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

Problem 2449

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

Problem 2450

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

Problem 2451

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

Problem 2452

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

Problem 2453

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

Problem 2454

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

Problem 2455

ODE	\[ \boxed {y^{\prime }+y^{3}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 3] \end {align*}
program solution	\[ \frac {1}{2 y^{2}} = t -\frac {17}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {3}{\sqrt {18 t -17}} \]

Problem 2456

ODE	\[ \boxed {y^{\prime }-\frac {{\mathrm e}^{t}}{y}=0} \] With initial conditions \begin {align*} [y \left (\ln \left (2\right )\right ) = -8] \end {align*}
program solution	\[ -{\mathrm e}^{t}+\frac {y^{2}}{2} = 30 \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\sqrt {2 \,{\mathrm e}^{t}+60} \]

Problem 2457

ODE	\[ \boxed {y^{\prime }=t \,{\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (1\right ) = 5] \end {align*}
program solution	\[ y = \frac {t \,{\mathrm e}^{2 t}}{2}-\frac {{\mathrm e}^{2 t}}{4}+5-\frac {{\mathrm e}^{2}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (2 t -1\right ) {\mathrm e}^{2 t}}{4}+5-\frac {{\mathrm e}^{2}}{4} \]

Problem 2458

ODE	\[ \boxed {y^{\prime }=\sin \left (t \right )^{2}} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{6}\right ) = 3\right ] \end {align*}
program solution	\[ y = \frac {t}{2}-\frac {\sin \left (2 t \right )}{4}-\frac {\pi }{12}+\frac {\sqrt {3}}{8}+3 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t}{2}+3-\frac {\pi }{12}+\frac {\sqrt {3}}{8}-\frac {\sin \left (2 t \right )}{4} \]

Problem 2459

ODE	\[ \boxed {y^{\prime }=8 \,{\mathrm e}^{4 t}+t} \] With initial conditions \begin {align*} [y \left (0\right ) = 12] \end {align*}
program solution	\[ y = \frac {t^{2}}{2}+2 \,{\mathrm e}^{4 t}+10 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{2}}{2}+2 \,{\mathrm e}^{4 t}+10 \]

Problem 2460

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

Problem 2461

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

Problem 2462

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

Problem 2463

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

Problem 2464

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

Problem 2465

ODE	\[ \boxed {y^{\prime }-y^{3}+y^{2}=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\textit {\_a}^{3}-\textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {1}{\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{t -1}\right )+1} \]

Problem 2466

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

Problem 2467

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

Problem 2468

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

Problem 2469

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

Problem 2470

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

Problem 2471

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

Problem 2472

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

Problem 2473

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

Problem 2474

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

Problem 2475

ODE	\[ \boxed {y^{\prime }-y \tan \left (t \right )=\sec \left (t \right )^{3}} \]
program solution	\[ y = \frac {\tan \left (t \right )+c_{1}}{\cos \left (t \right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sec \left (t \right ) \left (\tan \left (t \right )+c_{1} \right ) \]

Problem 2476

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

Problem 2477

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

Problem 2478

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

Problem 2479

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

Problem 2480

ODE	\[ \boxed {y^{\prime }-\frac {2 y}{t +1}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 6] \end {align*}
program solution	\[ y = 6 t^{2}+12 t +6 \] Verified OK.
Maple solution	\[ y \left (t \right ) = 6 \left (t +1\right )^{2} \]

Problem 2481

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

Problem 2482

ODE	\[ \boxed {y^{\prime }+4 \tan \left (2 t \right ) y=\tan \left (2 t \right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{8}\right ) = 2\right ] \end {align*}
program solution	\[ y = \frac {1}{4}+\frac {7 \cos \left (2 t \right )^{2}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2+\frac {7 \cos \left (4 t \right )}{4} \]

Problem 2483

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

Problem 2484

ODE	\[ \boxed {y^{\prime }-\frac {2 y}{-t^{2}+1}=3} \] With initial conditions \begin {align*} \left [y \left (\frac {1}{2}\right ) = 1\right ] \end {align*}
program solution	\[ y = \frac {-36 \ln \left (t +1\right ) t +36 \ln \left (3\right ) t -36 \ln \left (2\right ) t +18 t^{2}-36 \ln \left (t +1\right )+36 \ln \left (3\right )-36 \ln \left (2\right )+7 t -11}{6 t -6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (t +1\right ) \left (18 t -36 \ln \left (t +1\right )-11+36 \ln \left (3\right )-36 \ln \left (2\right )\right )}{6 t -6} \]

Problem 2485

ODE	\[ \boxed {y^{\prime }+\cot \left (t \right ) y=6 \cos \left (t \right )^{2}} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = 3\right ] \end {align*}
program solution	\[ y = -2 \csc \left (t \right ) \cos \left (t \right )^{3}+2 \csc \left (t \right ) \sqrt {2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -2 \csc \left (t \right ) \left (\cos \left (t \right )^{3}-\sqrt {2}\right ) \]

Problem 2486

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

Problem 2487

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

Problem 2488

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

Problem 2489

ODE	\[ \boxed {y \left (2 x^{2} y^{2}+1\right ) y^{\prime }+x \left (y^{4}+1\right )=0} \]
program solution	\[ \frac {x^{2} \left (y^{4}+1\right )}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-2-2 \sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\ y \left (x \right ) &= \frac {\sqrt {-2-2 \sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-1+\sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-1+\sqrt {-4 x^{4}-8 c_{1} x^{2}+1}}}{2 x} \\ \end{align*}

Problem 2490

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

Problem 2491

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

Problem 2492

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

Problem 2493

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

Problem 2494

ODE	\[ \boxed {\left (x +y^{3}\right ) y^{\prime }-y=0} \]
program solution	\[ \frac {y^{2}}{2}-\frac {x}{y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}-6 c_{1}}{3 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 i \sqrt {3}\, c_{1} +\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}-6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 i \sqrt {3}\, c_{1} -\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 2495

ODE	\[ \boxed {y^{\prime }+\frac {2 x^{2}+y^{2}+x}{y x}=0} \]
program solution	\[ \frac {x^{2} y^{2}}{2}+\frac {x^{4}}{2}+\frac {x^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-9 x^{4}-6 x^{3}+9 c_{1}}}{3 x} \\ y \left (x \right ) &= \frac {\sqrt {-9 x^{4}-6 x^{3}+9 c_{1}}}{3 x} \\ \end{align*}

Problem 2496

ODE	\[ \boxed {\left (y-x \right ) y^{\prime }+3 y=-2 x} \]
program solution	\[ \frac {\ln \left (y^{2}+2 y x +2 x^{2}\right )}{2}-2 \arctan \left (\frac {y+x}{x}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (-1+\tan \left (\operatorname {RootOf}\left (-4 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x \right )+2 c_{1} \right )\right )\right ) \]

Problem 2497

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

Problem 2498

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

Problem 2499

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

Problem 2500

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