2.17.48 Problems 4701 to 4800

Problem 4701

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

Problem 4702

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (2 x^{2}+x \right ) y^{\prime }-4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {4 x}{5}+\frac {2 x^{2}}{5}-\frac {16 x^{3}}{105}+\frac {x^{4}}{21}-\frac {4 x^{5}}{315}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {4 x}{3}+\frac {2 x^{2}}{3}-\frac {2 x^{4}}{9}+\frac {8 x^{5}}{45}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {4}{5} x +\frac {2}{5} x^{2}-\frac {16}{105} x^{3}+\frac {1}{21} x^{4}-\frac {4}{315} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-144+192 x -96 x^{2}+32 x^{4}-\frac {128}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 4703

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	\[ y = c_{1} \left (1-\frac {2 x}{3}+\frac {x^{2}}{6}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-4 x +6 x^{2}-4 x^{3}+x^{4}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {2}{3} x +\frac {1}{6} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-2+8 x -12 x^{2}+8 x^{3}-2 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 4704

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

Problem 4705

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x^{2}+\left (-2+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {3 x}{4}+\frac {3 x^{2}}{10}-\frac {x^{3}}{12}+\frac {x^{4}}{56}-\frac {x^{5}}{320}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{3}}{6}+\frac {x^{4}}{8}-\frac {x^{5}}{20}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {3}{4} x +\frac {3}{10} x^{2}-\frac {1}{12} x^{3}+\frac {1}{56} x^{4}-\frac {1}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-2 x^{3}+\frac {3}{2} x^{4}-\frac {3}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 4706

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x^{2}+\left (-2+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {x}{4}+\frac {x^{2}}{20}+\frac {x^{3}}{120}+\frac {x^{4}}{840}+\frac {x^{5}}{6720}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{4} x +\frac {1}{20} x^{2}+\frac {1}{120} x^{3}+\frac {1}{840} x^{4}+\frac {1}{6720} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+12 x +6 x^{2}+2 x^{3}+\frac {1}{2} x^{4}+\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 4707

ODE	\[ \boxed {x^{2} \left (1-4 x \right ) y^{\prime \prime }+\left (\left (1-n \right ) x -\left (6-4 n \right ) x^{2}\right ) y^{\prime }+n \left (1-n \right ) x y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-n x +\frac {\left (n -3\right ) n \,x^{2}}{2}-\frac {\left (n -4\right ) \left (n -5\right ) n \,x^{3}}{6}+\frac {\left (n -6\right ) \left (n -7\right ) \left (n -5\right ) n \,x^{4}}{24}-\frac {\left (n -8\right ) \left (n -9\right ) \left (n -6\right ) \left (n -7\right ) n \,x^{5}}{120}+O\left (x^{6}\right )\right )+c_{2} x^{n} \left (1+n x +\frac {\left (n +3\right ) n \,x^{2}}{2}+\frac {n \left (n +5\right ) \left (n +4\right ) x^{3}}{6}+\frac {\left (n +6\right ) \left (n +7\right ) \left (n +5\right ) n \,x^{4}}{24}+\frac {n \left (n +7\right ) \left (n +6\right ) \left (n +9\right ) \left (n +8\right ) x^{5}}{120}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{n} \left (1+n x +\frac {1}{2} n \left (n +3\right ) x^{2}+\frac {1}{6} \left (n +5\right ) \left (n +4\right ) n x^{3}+\frac {1}{24} n \left (n +5\right ) \left (n +7\right ) \left (n +6\right ) x^{4}+\frac {1}{120} \left (n +9\right ) \left (n +8\right ) \left (n +7\right ) \left (n +6\right ) n x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-n x +\frac {1}{2} n \left (-3+n \right ) x^{2}-\frac {1}{6} \left (-4+n \right ) \left (-5+n \right ) n x^{3}+\frac {1}{24} n \left (-5+n \right ) \left (-6+n \right ) \left (n -7\right ) x^{4}-\frac {1}{120} \left (-6+n \right ) \left (n -7\right ) \left (n -8\right ) \left (n -9\right ) n x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 4708

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }+\left (x -9\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1-\frac {4 x}{7}+\frac {5 x^{2}}{28}-\frac {5 x^{3}}{126}+\frac {x^{4}}{144}-\frac {x^{5}}{990}+\frac {x^{6}}{7920}+O\left (x^{7}\right )\right )+\frac {c_{2} \left (1-\frac {2 x}{5}+\frac {x^{2}}{20}-\frac {x^{6}}{7200}+O\left (x^{7}\right )\right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1-\frac {4}{7} x +\frac {5}{28} x^{2}-\frac {5}{126} x^{3}+\frac {1}{144} x^{4}-\frac {1}{990} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-86400+34560 x -4320 x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

Problem 4709

ODE	\[ \boxed {\left (a^{2}+x^{2}\right ) y^{\prime \prime }+x y^{\prime }-n^{2} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {n^{2} x^{2}}{2 a^{2}}+\frac {x^{4} n^{4}}{24 a^{4}}-\frac {x^{4} n^{2}}{6 a^{4}}+\frac {x^{6} n^{6}}{720 a^{6}}-\frac {x^{6} n^{4}}{36 a^{6}}+\frac {4 x^{6} n^{2}}{45 a^{6}}\right ) y \left (0\right )+\left (x +\frac {x^{3} n^{2}}{6 a^{2}}-\frac {x^{3}}{6 a^{2}}+\frac {x^{5} n^{4}}{120 a^{4}}-\frac {x^{5} n^{2}}{12 a^{4}}+\frac {3 x^{5}}{40 a^{4}}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {n^{2} x^{2}}{2 a^{2}}+\frac {n^{2} \left (n^{2}-4\right ) x^{4}}{24 a^{4}}\right ) c_{1} +\left (x +\frac {\left (n^{2}-1\right ) x^{3}}{6 a^{2}}+\frac {\left (n^{2}-1\right ) \left (n^{2}-9\right ) x^{5}}{120 a^{4}}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {n^{2} x^{2}}{2 a^{2}}+\frac {n^{2} \left (n^{2}-4\right ) x^{4}}{24 a^{4}}\right ) y \left (0\right )+\left (x +\frac {\left (n^{2}-1\right ) x^{3}}{6 a^{2}}+\frac {\left (n^{4}-10 n^{2}+9\right ) x^{5}}{120 a^{4}}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 4710

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-x y^{\prime }+a^{2} y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} a^{2} x^{2}+\frac {1}{24} a^{4} x^{4}-\frac {1}{6} a^{2} x^{4}-\frac {1}{720} x^{6} a^{6}+\frac {1}{36} x^{6} a^{4}-\frac {4}{45} x^{6} a^{2}\right ) y \left (0\right )+\left (x -\frac {1}{6} a^{2} x^{3}+\frac {1}{6} x^{3}+\frac {1}{120} a^{4} x^{5}-\frac {1}{12} a^{2} x^{5}+\frac {3}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {a^{2} x^{2}}{2}+\left (\frac {1}{24} a^{4}-\frac {1}{6} a^{2}\right ) x^{4}\right ) c_{1} +\left (x +\left (-\frac {a^{2}}{6}+\frac {1}{6}\right ) x^{3}+\left (\frac {1}{120} a^{4}-\frac {1}{12} a^{2}+\frac {3}{40}\right ) x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{2} a^{2}}{2}+\frac {a^{2} \left (a^{2}-4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (a^{2}-1\right ) x^{3}}{6}+\frac {\left (a^{4}-10 a^{2}+9\right ) x^{5}}{120}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 4711

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-x +\frac {x^{2}}{4}-\frac {x^{3}}{36}+\frac {x^{4}}{576}-\frac {x^{5}}{14400}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-x +\frac {x^{2}}{4}-\frac {x^{3}}{36}+\frac {x^{4}}{576}-\frac {x^{5}}{14400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+2 x -\frac {3 x^{2}}{4}+\frac {11 x^{3}}{108}-\frac {25 x^{4}}{3456}+\frac {137 x^{5}}{432000}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {1}{36} x^{3}+\frac {1}{576} x^{4}-\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}+\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 4712

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

Problem 4713

ODE	\[ \boxed {x y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{2}+\frac {x^{2}}{12}-\frac {x^{3}}{144}+\frac {x^{4}}{2880}-\frac {x^{5}}{86400}+O\left (x^{6}\right )\right )+c_{2} \left (-x \left (1-\frac {x}{2}+\frac {x^{2}}{12}-\frac {x^{3}}{144}+\frac {x^{4}}{2880}-\frac {x^{5}}{86400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {3 x^{2}}{4}+\frac {7 x^{3}}{36}-\frac {35 x^{4}}{1728}+\frac {101 x^{5}}{86400}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}-\frac {1}{144} x^{3}+\frac {1}{2880} x^{4}-\frac {1}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}+\frac {1}{144} x^{4}-\frac {1}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {7}{36} x^{3}-\frac {35}{1728} x^{4}+\frac {101}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 4714

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

Problem 4715

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (3 x -1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {4 x}{3}+\frac {5 x^{2}}{6}-\frac {x^{3}}{3}+\frac {7 x^{4}}{72}-\frac {x^{5}}{45}+O\left (x^{6}\right )\right )+c_{2} \left (-3 x \left (1-\frac {4 x}{3}+\frac {5 x^{2}}{6}-\frac {x^{3}}{3}+\frac {7 x^{4}}{72}-\frac {x^{5}}{45}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1+2 x -\frac {13 x^{3}}{3}+\frac {49 x^{4}}{12}-2 x^{5}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {4}{3} x +\frac {5}{6} x^{2}-\frac {1}{3} x^{3}+\frac {7}{72} x^{4}-\frac {1}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (6 x^{2}-8 x^{3}+5 x^{4}-2 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-4 x +5 x^{2}+2 x^{3}-4 x^{4}+\frac {7}{3} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

Problem 4716

ODE	\[ \boxed {\left (-x^{2}+x \right ) y^{\prime \prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x}{2}+\frac {x^{2}}{4}+\frac {7 x^{3}}{48}+\frac {91 x^{4}}{960}+\frac {637 x^{5}}{9600}+O\left (x^{6}\right )\right )+c_{2} \left (x \left (1+\frac {x}{2}+\frac {x^{2}}{4}+\frac {7 x^{3}}{48}+\frac {91 x^{4}}{960}+\frac {637 x^{5}}{9600}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1-\frac {x^{2}}{4}-\frac {x^{3}}{12}-\frac {17 x^{4}}{576}-\frac {311 x^{5}}{28800}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+\frac {1}{2} x +\frac {1}{4} x^{2}+\frac {7}{48} x^{3}+\frac {91}{960} x^{4}+\frac {637}{9600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x +\frac {1}{2} x^{2}+\frac {1}{4} x^{3}+\frac {7}{48} x^{4}+\frac {91}{960} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {1}{4} x^{2}-\frac {1}{12} x^{3}-\frac {17}{576} x^{4}-\frac {311}{28800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 4717

ODE	\[ \boxed {x \left (-x^{2}+1\right ) y^{\prime \prime }+\left (-3 x^{2}+1\right ) y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{4}+\frac {9 x^{4}}{64}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1+\frac {x^{2}}{4}+\frac {9 x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{4}+\frac {21 x^{4}}{128}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{4} x^{2}+\frac {9}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}+\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 4718

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

Problem 4719

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (x^{2}+4 x \right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right )+c_{2} \left (\frac {x^{4} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+x \left (1-\frac {x}{2}+\frac {x^{2}}{2}-\frac {x^{4}}{2}-\frac {3 x^{5}}{8}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x^{3} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (6 x^{3}+6 x^{4}+3 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12-6 x +6 x^{2}+11 x^{3}+5 x^{4}+x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )\right ) x \]

Problem 4720

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

Problem 4721

ODE	\[ \boxed {4 x \left (1-x \right ) y^{\prime \prime }-4 y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {3 x}{4}+\frac {75 x^{2}}{128}+\frac {245 x^{3}}{512}+\frac {6615 x^{4}}{16384}+\frac {22869 x^{5}}{65536}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {x^{2} \left (1+\frac {3 x}{4}+\frac {75 x^{2}}{128}+\frac {245 x^{3}}{512}+\frac {6615 x^{4}}{16384}+\frac {22869 x^{5}}{65536}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{32}+1-\frac {x}{4}-\frac {15 x^{4}}{16384}-\frac {259 x^{5}}{196608}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {3}{4} x +\frac {75}{128} x^{2}+\frac {245}{512} x^{3}+\frac {6615}{16384} x^{4}+\frac {22869}{65536} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{16} x^{2}+\frac {3}{64} x^{3}+\frac {75}{2048} x^{4}+\frac {245}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {1}{2} x +\frac {1}{2} x^{2}+\frac {3}{8} x^{3}+\frac {2415}{8192} x^{4}+\frac {23779}{98304} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 4722

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

Problem 4723

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

Problem 4724

ODE	\[ \boxed {\left (-x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }+2 y=3 x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{3}}{5}+\frac {x^{4}}{30}+\frac {x^{5}}{105}+O\left (x^{6}\right )+c_{1} \left (1-\frac {2 x}{3}+\frac {x^{2}}{6}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-4 x +6 x^{2}-4 x^{3}+x^{4}+O\left (x^{6}\right )\right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {2}{3} x +\frac {1}{6} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-2+8 x -12 x^{2}+8 x^{3}-2 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}}+x^{3} \left (\frac {1}{5}+\frac {1}{30} x +\frac {1}{105} x^{2}+\operatorname {O}\left (x^{3}\right )\right ) \]

Problem 4725

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x}{6}+\frac {3 x^{2}}{40}+\frac {5 x^{3}}{112}+\frac {35 x^{4}}{1152}+\frac {63 x^{5}}{2816}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} \left (1+\frac {1}{6} x +\frac {3}{40} x^{2}+\frac {5}{112} x^{3}+\frac {35}{1152} x^{4}+\frac {63}{2816} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 4726

ODE	\[ \boxed {2 x \left (1-x \right ) y^{\prime \prime }+y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+O\left (x^{6}\right )\right )+c_{2} \left (\frac {x \left (1+O\left (x^{6}\right )\right ) \ln \left (x \right )}{2}+1+\frac {x^{2}}{8}+\frac {x^{3}}{32}+\frac {5 x^{4}}{384}+\frac {7 x^{5}}{1024}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\frac {1}{2} x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_{2} +c_{1} x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{32} x^{3}+\frac {5}{384} x^{4}+\frac {7}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

Problem 4727

ODE	\[ \boxed {2 x \left (1-x \right ) y^{\prime \prime }+\left (1-11 x \right ) y^{\prime }-10 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+5 x +14 x^{2}+30 x^{3}+55 x^{4}+91 x^{5}+O\left (x^{6}\right )\right )+c_{2} \left (1+10 x +35 x^{2}+84 x^{3}+165 x^{4}+286 x^{5}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+5 x +14 x^{2}+30 x^{3}+55 x^{4}+91 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+10 x +35 x^{2}+84 x^{3}+165 x^{4}+286 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 4728

ODE	\[ \boxed {x \left (1-x \right ) y^{\prime \prime }+\frac {\left (-2 x +1\right ) y^{\prime }}{3}+\frac {20 y}{9}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1-\frac {6 x}{5}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {20 x}{3}+\frac {35 x^{2}}{9}+\frac {50 x^{3}}{81}+\frac {65 x^{4}}{243}+\frac {112 x^{5}}{729}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {2}{3}} \left (1-\frac {6}{5} x +\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {20}{3} x +\frac {35}{9} x^{2}+\frac {50}{81} x^{3}+\frac {65}{243} x^{4}+\frac {112}{729} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 4729

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

Problem 4730

ODE	\[ \boxed {4 y^{\prime \prime }+\frac {3 \left (-x^{2}+2\right ) y}{\left (-x^{2}+1\right )^{2}}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{4} x^{2}-\frac {3}{32} x^{4}-\frac {5}{128} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}-\frac {3}{32} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{4} x^{2}-\frac {3}{32} x^{4}\right ) c_{1} +\left (x -\frac {1}{4} x^{3}-\frac {3}{32} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{4} x^{2}-\frac {3}{32} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}-\frac {3}{32} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 4731

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

Problem 4732

ODE	\[ \boxed {u^{\prime \prime }-\frac {a^{2} u}{x^{\frac {2}{3}}}=0} \]
program solution	\[ u = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {3}{4}, \frac {3 i a \,x^{\frac {2}{3}}}{2}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {3}{4}, \frac {3 i a \,x^{\frac {2}{3}}}{2}\right ) \] Verified OK.
Maple solution	\[ u \left (x \right ) = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {3}{4}, \frac {3 \sqrt {-a^{2}}\, x^{\frac {2}{3}}}{2}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {3}{4}, \frac {3 \sqrt {-a^{2}}\, x^{\frac {2}{3}}}{2}\right ) c_{1} \right ) \]

Problem 4733

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

Problem 4734

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

Problem 4735

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

Problem 4736

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

Problem 4737

ODE	\[ \boxed {u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u=0} \]
program solution	\[ u = \frac {c_{1} \left (a x -i\right ) {\mathrm e}^{-i a x}}{x^{3} a}-\frac {c_{2} \left (i a x -1\right ) {\mathrm e}^{i a x}}{2 a^{2} x^{3}} \] Verified OK.
Maple solution	\[ u \left (x \right ) = \frac {\left (a c_{1} x +c_{2} \right ) \cos \left (a x \right )+\sin \left (a x \right ) \left (a c_{2} x -c_{1} \right )}{x^{3}} \]

Problem 4738

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

Problem 4739

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

Problem 4740

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

Problem 4741

ODE	\[ \boxed {y^{\prime \prime } x^{2}+y^{\prime } x +\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}}=0} \]
program solution	\[ y = \frac {c_{1} \left (-i a +x \right ) {\mathrm e}^{-\frac {i x}{a}}}{x^{\frac {3}{2}}}-\frac {c_{2} \left (i x -a \right ) a \,{\mathrm e}^{\frac {i x}{a}}}{2 x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (i x +a \right ) c_{2} {\mathrm e}^{-\frac {i x}{a}}+\left (-i x +a \right ) {\mathrm e}^{\frac {i x}{a}} c_{1}}{x^{\frac {3}{2}}} \]

Problem 4742

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}-3 i x -3\right ) {\mathrm e}^{-i x}}{x^{\frac {5}{2}}}-\frac {c_{2} {\mathrm e}^{i x} \left (i x^{2}-3 x -3 i\right )}{2 x^{\frac {5}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-3 \left (i x -\frac {1}{3} x^{2}+1\right ) c_{2} {\mathrm e}^{-i x}+3 \left (i x +\frac {1}{3} x^{2}-1\right ) c_{1} {\mathrm e}^{i x}}{x^{\frac {5}{2}}} \]

Problem 4743

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

Problem 4744

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

Problem 4745

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

Problem 4746

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

Problem 4747

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

Problem 4748

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

Problem 4749

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

Problem 4750

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

Problem 4751

ODE	\[ \boxed {y^{\prime } \sin \left (x \right )-y \ln \left (y\right )=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{3}\right ) = {\mathrm e}\right ] \end {align*}
program solution	\[ y = {\mathrm e}^{-\sqrt {3}\, \left (-\csc \left (x \right )+\cot \left (x \right )\right )} \] Verified OK. {positive}
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\left (\cot \left (x \right )-\csc \left (x \right )\right ) \sqrt {3}} \]

Problem 4752

ODE	\[ \boxed {y^{2}+x y y^{\prime }=-1} \] With initial conditions \begin {align*} [y \left (5\right ) = 0] \end {align*}
program solution	\[ \sqrt {y^{2}+1} = \frac {5}{x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\sqrt {-x^{2}+25}}{x} \\ y \left (x \right ) &= -\frac {\sqrt {-x^{2}+25}}{x} \\ \end{align*}

Problem 4753

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

Problem 4754

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

Problem 4755

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

Problem 4756

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

Problem 4757

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

Problem 4758

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

Problem 4759

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

Problem 4760

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

Problem 4761

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

Problem 4762

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

Problem 4763

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

Problem 4764

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

Problem 4765

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

Problem 4766

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

Problem 4767

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

Problem 4768

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

Problem 4769

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

Problem 4770

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

Problem 4771

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

Problem 4772

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

Problem 4773

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

Problem 4774

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

Problem 4775

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

Problem 4776

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

Problem 4777

ODE	\[ \boxed {3 x y^{2} y^{\prime }+3 y^{3}=1} \]
program solution	\[ -\ln \left (x \right )-\frac {\ln \left (3 y^{3}-1\right )}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (9 x^{3}+27 c_{1} \right )^{\frac {1}{3}}}{3 x} \\ y \left (x \right ) &= -\frac {\left (9 x^{3}+27 c_{1} \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{6 x} \\ y \left (x \right ) &= \frac {\left (9 x^{3}+27 c_{1} \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{6 x} \\ \end{align*}

Problem 4778

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

Problem 4779

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

Problem 4780

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

Problem 4781

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

Problem 4782

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

Problem 4783

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

Problem 4784

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

Problem 4785

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

Problem 4786

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

Problem 4787

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

Problem 4788

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

Problem 4789

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

Problem 4790

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

Problem 4791

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

Problem 4792

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

Problem 4793

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

Problem 4794

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 4795

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

Problem 4796

ODE	\[ \boxed {y^{\prime \prime }+16 y=0} \]
program solution	\[ y = c_{1} \cos \left (4 x \right )+\frac {c_{2} \sin \left (4 x \right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (4 x \right )+c_{2} \cos \left (4 x \right ) \]

Problem 4797

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

Problem 4798

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

Problem 4799

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+13 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x} \cos \left (3 x \right )+\frac {c_{2} {\mathrm e}^{2 x} \sin \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right )\right ) \]

Problem 4800

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