Problem 14001
| ODE |
\[ \boxed {y^{\prime \prime }+\frac {2 y^{\prime }}{x +2}+y=0} \] With the expansion point for the power series method at \(x = -2\). |
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program solution |
\[ y = c_{1} \left (1-\frac {\left (x +2\right )^{2}}{6}+\frac {\left (x +2\right )^{4}}{120}+O\left (\left (x +2\right )^{6}\right )\right )+\frac {c_{2} \left (1-\frac {\left (x +2\right )^{2}}{2}+\frac {\left (x +2\right )^{4}}{24}+O\left (\left (x +2\right )^{6}\right )\right )}{x +2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (1-\frac {1}{6} \left (x +2\right )^{2}+\frac {1}{120} \left (x +2\right )^{4}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} \left (x +2\right )^{2}+\frac {1}{24} \left (x +2\right )^{4}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )}{x +2} \] |
Problem 14002
| ODE |
\[ \boxed {4 y^{\prime \prime }+\frac {\left (4 x -3\right ) y}{\left (x -1\right )^{2}}=0} \] With the expansion point for the power series method at \(x = 1\). |
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program solution |
\[ y = c_{1} \sqrt {x -1}\, \left (2-x +\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{36}+\frac {\left (x -1\right )^{4}}{576}-\frac {\left (x -1\right )^{5}}{14400}+O\left (\left (x -1\right )^{6}\right )\right )+c_{2} \left (\sqrt {x -1}\, \left (2-x +\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{36}+\frac {\left (x -1\right )^{4}}{576}-\frac {\left (x -1\right )^{5}}{14400}+O\left (\left (x -1\right )^{6}\right )\right ) \ln \left (x -1\right )+\sqrt {x -1}\, \left (2 x -2-\frac {3 \left (x -1\right )^{2}}{4}+\frac {11 \left (x -1\right )^{3}}{108}-\frac {25 \left (x -1\right )^{4}}{3456}+\frac {137 \left (x -1\right )^{5}}{432000}+O\left (\left (x -1\right )^{6}\right )\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\left (\ln \left (-1+x \right ) c_{2} +c_{1} \right ) \left (1-\left (-1+x \right )+\frac {1}{4} \left (-1+x \right )^{2}-\frac {1}{36} \left (-1+x \right )^{3}+\frac {1}{576} \left (-1+x \right )^{4}-\frac {1}{14400} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right )+\left (2 \left (-1+x \right )-\frac {3}{4} \left (-1+x \right )^{2}+\frac {11}{108} \left (-1+x \right )^{3}-\frac {25}{3456} \left (-1+x \right )^{4}+\frac {137}{432000} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right ) c_{2} \right ) \sqrt {-1+x} \] |
Problem 14003
| ODE |
\[ \boxed {\left (x -3\right )^{2} y^{\prime \prime }+\left (x^{2}-3 x \right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 3\). |
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program solution |
\[ y = c_{1} \left (x -3\right ) \left (\frac {8}{5}-\frac {x}{5}+\frac {\left (x -3\right )^{2}}{30}-\frac {\left (x -3\right )^{3}}{210}+\frac {\left (x -3\right )^{4}}{1680}-\frac {\left (x -3\right )^{5}}{15120}+O\left (\left (x -3\right )^{6}\right )\right )+\frac {c_{2} \left (4-x +\frac {\left (x -3\right )^{2}}{2}-\frac {\left (x -3\right )^{3}}{6}+\frac {\left (x -3\right )^{4}}{24}-\frac {\left (x -3\right )^{5}}{120}+O\left (\left (x -3\right )^{6}\right )\right )}{\left (x -3\right )^{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (-3+x \right )^{4} \left (1-\frac {1}{5} \left (-3+x \right )+\frac {1}{30} \left (-3+x \right )^{2}-\frac {1}{210} \left (-3+x \right )^{3}+\frac {1}{1680} \left (-3+x \right )^{4}-\frac {1}{15120} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )+c_{2} \left (-144+144 \left (-3+x \right )-72 \left (-3+x \right )^{2}+24 \left (-3+x \right )^{3}-6 \left (-3+x \right )^{4}+\frac {6}{5} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )}{\left (-3+x \right )^{3}} \] |
Problem 14004
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (-x^{2}+2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} x^{2} \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{120}+O\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{24}+O\left (x^{6}\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x^{2} \left (1+\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \] |
Problem 14005
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-2 x^{2} y^{\prime }+\left (x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} x^{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{3}+\frac {5 x^{4}}{24}+\frac {11 x^{5}}{120}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x^{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+12 x +6 x^{2}+4 x^{3}+\frac {5}{2} x^{4}+\frac {11}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \] |
Problem 14006
| ODE |
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{4}-\frac {3 x^{4}}{128}+O\left (x^{6}\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \] |
Problem 14007
| ODE |
\[ \boxed {x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+\left (4 x^{2}+5 x \right ) y^{\prime }+y \left (x^{2}+1\right )=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \frac {c_{1} \left (1+\frac {2 x}{3}-\frac {19 x^{2}}{120}+\frac {x^{3}}{180}-\frac {23 x^{4}}{51840}+\frac {557 x^{5}}{1425600}+O\left (x^{6}\right )\right )}{\sqrt {x}}+\frac {c_{2} \left (1+4 x +\frac {x^{2}}{6}-\frac {14 x^{3}}{45}+\frac {209 x^{4}}{2520}-\frac {823 x^{5}}{28350}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (1+4 x +\frac {1}{6} x^{2}-\frac {14}{45} x^{3}+\frac {209}{2520} x^{4}-\frac {823}{28350} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1+\frac {2}{3} x -\frac {19}{120} x^{2}+\frac {1}{180} x^{3}-\frac {23}{51840} x^{4}+\frac {557}{1425600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \] |
Problem 14008
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-\left (2 x^{2}+5 x \right ) y^{\prime }+9 y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} x^{3} \left (12 x^{2}+6 x +1+\frac {40 x^{3}}{3}+10 x^{4}+\frac {28 x^{5}}{5}+O\left (x^{6}\right )\right )+c_{2} \left (x^{3} \left (12 x^{2}+6 x +1+\frac {40 x^{3}}{3}+10 x^{4}+\frac {28 x^{5}}{5}+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{3} \left (-29 x^{2}-10 x -\frac {346 x^{3}}{9}-\frac {193 x^{4}}{6}-\frac {1459 x^{5}}{75}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+6 x +12 x^{2}+\frac {40}{3} x^{3}+10 x^{4}+\frac {28}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-10\right ) x -29 x^{2}-\frac {346}{9} x^{3}-\frac {193}{6} x^{4}-\frac {1459}{75} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{3} \] |
Problem 14009
| ODE |
\[ \boxed {x^{2} \left (2 x +1\right ) y^{\prime \prime }+y^{\prime } x +\left (4 x^{3}-4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} x^{2} \left (1-\frac {4 x}{5}+\frac {4 x^{2}}{5}-\frac {116 x^{3}}{105}+\frac {311 x^{4}}{210}-\frac {358 x^{5}}{175}+O\left (x^{6}\right )\right )+c_{2} \left (-4 x^{2} \left (1-\frac {4 x}{5}+\frac {4 x^{2}}{5}-\frac {116 x^{3}}{105}+\frac {311 x^{4}}{210}-\frac {358 x^{5}}{175}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1+4 x +4 x^{2}+\frac {4 x^{3}}{3}-\frac {56 x^{5}}{25}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {4}{5} x +\frac {4}{5} x^{2}-\frac {116}{105} x^{3}+\frac {311}{210} x^{4}-\frac {358}{175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (576 x^{4}-\frac {2304}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-576 x -576 x^{2}-192 x^{3}-384 x^{4}+\frac {15744}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \] |
Problem 14010
| ODE |
\[ \boxed {4 x^{2} y^{\prime \prime }+8 y^{\prime } x +\left (1-4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = \frac {c_{1} \left (x +1+\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {x^{4}}{576}+\frac {x^{5}}{14400}+O\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} \left (\frac {\left (x +1+\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {x^{4}}{576}+\frac {x^{5}}{14400}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{\sqrt {x}}+\frac {-2 x -\frac {3 x^{2}}{4}-\frac {11 x^{3}}{108}-\frac {25 x^{4}}{3456}-\frac {137 x^{5}}{432000}+O\left (x^{6}\right )}{\sqrt {x}}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {1}{576} x^{4}+\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -\frac {3}{4} x^{2}-\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}-\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{\sqrt {x}} \] |
Problem 14011
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -\left (2 x +1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} x \left (1+\frac {2 x}{3}+\frac {x^{2}}{6}+\frac {x^{3}}{45}+\frac {x^{4}}{540}+\frac {x^{5}}{9450}+O\left (x^{6}\right )\right )+c_{2} \left (-2 x \left (1+\frac {2 x}{3}+\frac {x^{2}}{6}+\frac {x^{3}}{45}+\frac {x^{4}}{540}+\frac {x^{5}}{9450}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1-2 x +\frac {16 x^{3}}{9}+\frac {25 x^{4}}{36}+\frac {157 x^{5}}{1350}+O\left (x^{6}\right )}{x}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1+\frac {2}{3} x +\frac {1}{6} x^{2}+\frac {1}{45} x^{3}+\frac {1}{540} x^{4}+\frac {1}{9450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (4 x^{2}+\frac {8}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{45} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+4 x -\frac {32}{9} x^{3}-\frac {25}{18} x^{4}-\frac {157}{675} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \] |
Problem 14012
| ODE |
\[ \boxed {x y^{\prime \prime }+4 y^{\prime }+\frac {12 y}{\left (x +2\right )^{2}}=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} \left (1-\frac {3 x}{4}+\frac {21 x^{2}}{40}-\frac {27 x^{3}}{80}+\frac {33 x^{4}}{160}-\frac {39 x^{5}}{320}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {3 x}{2}+\frac {3 x^{2}}{4}+\frac {x^{3}}{8}+O\left (x^{6}\right )\right )}{x^{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (1-\frac {3}{4} x +\frac {21}{40} x^{2}-\frac {27}{80} x^{3}+\frac {33}{160} x^{4}-\frac {39}{320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12+18 x +9 x^{2}+\frac {9}{8} x^{4}-\frac {63}{80} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \] |
Problem 14013
| ODE |
\[ \boxed {x y^{\prime \prime }+4 y^{\prime }+\frac {12 y}{\left (x +2\right )^{2}}=0} \] With the expansion point for the power series method at \(x = -2\). |
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program solution |
\[ y = c_{1} \left (x +2\right )^{3} \left (4+\frac {3 x}{2}+\frac {3 \left (x +2\right )^{2}}{2}+\frac {5 \left (x +2\right )^{3}}{4}+\frac {15 \left (x +2\right )^{4}}{16}+\frac {21 \left (x +2\right )^{5}}{32}+O\left (\left (x +2\right )^{6}\right )\right )+\frac {c_{2} \left (\frac {3}{2}+\frac {x}{4}+\frac {\left (x +2\right )^{2}}{24}+\frac {\left (x +2\right )^{5}}{192}+O\left (\left (x +2\right )^{6}\right )\right )}{\left (x +2\right )^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x +2\right )^{5} \left (1+\frac {3}{2} \left (x +2\right )+\frac {3}{2} \left (x +2\right )^{2}+\frac {5}{4} \left (x +2\right )^{3}+\frac {15}{16} \left (x +2\right )^{4}+\frac {21}{32} \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )+c_{2} \left (2880+720 \left (x +2\right )+120 \left (x +2\right )^{2}+15 \left (x +2\right )^{5}+\operatorname {O}\left (\left (x +2\right )^{6}\right )\right )}{\left (x +2\right )^{2}} \] |
Problem 14014
| ODE |
\[ \boxed {\left (x -3\right ) y^{\prime \prime }+\left (x -3\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 3\). |
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program solution |
\[ y = c_{1} \left (x -3\right ) \left (4-x +\frac {\left (x -3\right )^{2}}{2}-\frac {\left (x -3\right )^{3}}{6}+\frac {\left (x -3\right )^{4}}{24}-\frac {\left (x -3\right )^{5}}{120}+O\left (\left (x -3\right )^{6}\right )\right )+c_{2} \left (-\left (x -3\right ) \left (4-x +\frac {\left (x -3\right )^{2}}{2}-\frac {\left (x -3\right )^{3}}{6}+\frac {\left (x -3\right )^{4}}{24}-\frac {\left (x -3\right )^{5}}{120}+O\left (\left (x -3\right )^{6}\right )\right ) \ln \left (x -3\right )+1-\left (x -3\right )^{2}+\frac {3 \left (x -3\right )^{3}}{4}-\frac {11 \left (x -3\right )^{4}}{36}+\frac {25 \left (x -3\right )^{5}}{288}+O\left (\left (x -3\right )^{6}\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \left (-3+x \right ) \left (1-\left (-3+x \right )+\frac {1}{2} \left (-3+x \right )^{2}-\frac {1}{6} \left (-3+x \right )^{3}+\frac {1}{24} \left (-3+x \right )^{4}-\frac {1}{120} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )+c_{2} \left (\ln \left (-3+x \right ) \left (-\left (-3+x \right )+\left (-3+x \right )^{2}-\frac {1}{2} \left (-3+x \right )^{3}+\frac {1}{6} \left (-3+x \right )^{4}-\frac {1}{24} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )+\left (1-\left (-3+x \right )+\frac {1}{4} \left (-3+x \right )^{3}-\frac {5}{36} \left (-3+x \right )^{4}+\frac {13}{288} \left (-3+x \right )^{5}+\operatorname {O}\left (\left (-3+x \right )^{6}\right )\right )\right ) \] |
Problem 14015
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +3 y=0} \] With the expansion point for the power series method at \(x = 1\). |
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program solution |
\[ y = c_{1} \sqrt {x -1}\, \left (\frac {1}{12}+\frac {11 x}{12}+\frac {11 \left (x -1\right )^{2}}{160}-\frac {143 \left (x -1\right )^{3}}{13440}+\frac {5291 \left (x -1\right )^{4}}{1935360}-\frac {11063 \left (x -1\right )^{5}}{12902400}+O\left (\left (x -1\right )^{6}\right )\right )+c_{2} \left (-2+3 x +\left (x -1\right )^{2}-\frac {\left (x -1\right )^{3}}{15}+\frac {\left (x -1\right )^{4}}{70}-\frac {13 \left (x -1\right )^{5}}{3150}+O\left (\left (x -1\right )^{6}\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} \sqrt {-1+x}\, \left (1+\frac {11}{12} \left (-1+x \right )+\frac {11}{160} \left (-1+x \right )^{2}-\frac {143}{13440} \left (-1+x \right )^{3}+\frac {5291}{1935360} \left (-1+x \right )^{4}-\frac {11063}{12902400} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right )+c_{2} \left (1+3 \left (-1+x \right )+\left (-1+x \right )^{2}-\frac {1}{15} \left (-1+x \right )^{3}+\frac {1}{70} \left (-1+x \right )^{4}-\frac {13}{3150} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right ) \] |
Problem 14016
| ODE |
\[ \boxed {4 x^{2} y^{\prime \prime }+\left (1-4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} \sqrt {x}\, \left (x +1+\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {x^{4}}{576}+\frac {x^{5}}{14400}+O\left (x^{6}\right )\right )+c_{2} \left (\sqrt {x}\, \left (x +1+\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {x^{4}}{576}+\frac {x^{5}}{14400}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-2 x -\frac {3 x^{2}}{4}-\frac {11 x^{3}}{108}-\frac {25 x^{4}}{3456}-\frac {137 x^{5}}{432000}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1+x +\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {1}{576} x^{4}+\frac {1}{14400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -\frac {3}{4} x^{2}-\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}-\frac {137}{432000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \sqrt {x} \] |
Problem 14017
| ODE |
\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+y=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} \left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right )+c_{2} \left (\left (1-\frac {x^{2}}{4}+\frac {x^{4}}{64}+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {x^{2}}{4}-\frac {3 x^{4}}{128}+O\left (x^{6}\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \] |
Problem 14018
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-\left (x^{2}+x \right ) y^{\prime }+4 y x=0} \] With the expansion point for the power series method at \(x = 0\). |
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program solution |
\[ y = c_{1} x^{2} \left (1-\frac {2 x}{3}+\frac {x^{2}}{12}+O\left (x^{6}\right )\right )+c_{2} \left (-6 x^{2} \left (1-\frac {2 x}{3}+\frac {x^{2}}{12}+O\left (x^{6}\right )\right ) \ln \left (x \right )+1+4 x -\frac {22 x^{3}}{3}+\frac {43 x^{4}}{24}-\frac {x^{5}}{30}+O\left (x^{6}\right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x^{2} \left (1-\frac {2}{3} x +\frac {1}{12} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\ln \left (x \right ) \left (12 x^{2}-8 x^{3}+x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (-2-8 x -7 x^{2}+\frac {58}{3} x^{3}-\frac {25}{6} x^{4}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \] |
Problem 14019
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (4 x -4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\). |
|
program solution |
\[ y = c_{1} x^{2} \left (1-\frac {4 x}{5}+\frac {4 x^{2}}{15}-\frac {16 x^{3}}{315}+\frac {2 x^{4}}{315}-\frac {8 x^{5}}{14175}+O\left (x^{6}\right )\right )+c_{2} \left (-\frac {16 x^{2} \left (1-\frac {4 x}{5}+\frac {4 x^{2}}{15}-\frac {16 x^{3}}{315}+\frac {2 x^{4}}{315}-\frac {8 x^{5}}{14175}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{9}+\frac {1+\frac {4 x}{3}+\frac {4 x^{2}}{3}+\frac {16 x^{3}}{9}-\frac {128 x^{5}}{75}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1-\frac {4}{5} x +\frac {4}{15} x^{2}-\frac {16}{315} x^{3}+\frac {2}{315} x^{4}-\frac {8}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (256 x^{4}-\frac {1024}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-192 x -192 x^{2}-256 x^{3}+\frac {6144}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \] |
Problem 14020
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 y \left (t \right )\\ y^{\prime }\left (t \right )&=1-2 x \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{2} \sin \left (2 t \right )+c_{1} \cos \left (2 t \right )+\frac {1}{2} \\ y \left (t \right ) &= c_{2} \cos \left (2 t \right )-c_{1} \sin \left (2 t \right ) \\ \end{align*} |
Problem 14021
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=6 x \left (t \right )-7 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-5 t}+c_{2} {\mathrm e}^{2 t} \\ y \left (t \right ) &= 3 c_{1} {\mathrm e}^{-5 t}+\frac {2 c_{2} {\mathrm e}^{2 t}}{3} \\ \end{align*} |
Problem 14022
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=\frac {15 y \left (t \right )}{t}-\frac {2 x \left (t \right )}{t}\\ y^{\prime }\left (t \right )&=\frac {x \left (t \right )}{t} \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {3 c_{1} t^{8}-5 c_{2}}{t^{5}} \\ y \left (t \right ) &= \frac {c_{1} t^{8}+c_{2}}{t^{5}} \\ \end{align*} |
Problem 14023
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=5 x \left (t \right )-2 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 7, y \left (0\right ) = -7] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{3 t}+4 \,{\mathrm e}^{-4 t} \\ y \left (t \right ) &= 3 \,{\mathrm e}^{3 t}-10 \,{\mathrm e}^{-4 t} \\ \end{align*} |
Problem 14024
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+4 y \left (t \right )\\ y^{\prime }\left (t \right )&=8 x \left (t \right )+y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 9] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -3 \,{\mathrm e}^{-3 t}+3 \,{\mathrm e}^{9 t} \\ y \left (t \right ) &= 6 \,{\mathrm e}^{-3 t}+3 \,{\mathrm e}^{9 t} \\ \end{align*} |
Problem 14025
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = -21] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 6 \,{\mathrm e}^{-2 t}-6 \,{\mathrm e}^{5 t} \\ y \left (t \right ) &= -18 \,{\mathrm e}^{-2 t}-3 \,{\mathrm e}^{5 t} \\ \end{align*} |
Problem 14026
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=5 x \left (t \right )-2 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 1, y \left (0\right ) = 15] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 5 \,{\mathrm e}^{3 t}-4 \,{\mathrm e}^{-4 t} \\ y \left (t \right ) &= 5 \,{\mathrm e}^{3 t}+10 \,{\mathrm e}^{-4 t} \\ \end{align*} |
Problem 14027
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{2 t}-c_{2} {\mathrm e}^{-2 t} \\ \end{align*} |
Problem 14028
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right ) \\ y \left (t \right ) &= c_{1} \cos \left (2 t \right )-c_{2} \sin \left (2 t \right ) \\ \end{align*} |
Problem 14029
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=8 x \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} \sin \left (4 t \right )+c_{2} \cos \left (4 t \right ) \\ y \left (t \right ) &= -2 c_{1} \cos \left (4 t \right )+2 c_{2} \sin \left (4 t \right ) \\ \end{align*} |
Problem 14030
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-13 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 2, y \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (-3 \sin \left (3 t \right )+2 \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{2 t} \cos \left (3 t \right ) \\ \end{align*} |
Problem 14031
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )+3 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = a_{1}, y \left (0\right ) = a_{2}] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (a_{2} \sin \left (2 t \right )+a_{1} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{3 t} \left (a_{2} \cos \left (2 t \right )-a_{1} \sin \left (2 t \right )\right ) \\ \end{align*} |
Problem 14032
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=8 x \left (t \right )+2 y \left (t \right )-17\\ y^{\prime }\left (t \right )&=4 x \left (t \right )+y \left (t \right )-13 \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0] \] |
|
program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -2 \,{\mathrm e}^{9 t}+t +2 \\ y \left (t \right ) &= -{\mathrm e}^{9 t}+1-4 t \\ \end{align*} |
Problem 14033
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=8 x \left (t \right )+2 y \left (t \right )+7 \,{\mathrm e}^{2 t}\\ y^{\prime }\left (t \right )&=4 x \left (t \right )+y \left (t \right )-7 \,{\mathrm e}^{2 t} \end {align*} With initial conditions \[ [x \left (0\right ) = -1, y \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -\frac {3}{2}+\frac {{\mathrm e}^{2 t}}{2} \\ y \left (t \right ) &= 6-5 \,{\mathrm e}^{2 t} \\ \end{align*} |
Problem 14034
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+3 y \left (t \right )-6 \,{\mathrm e}^{3 t}\\ y^{\prime }\left (t \right )&=x \left (t \right )+6 y \left (t \right )+2 \,{\mathrm e}^{3 t} \end {align*} With initial conditions \[ [x \left (0\right ) = 4, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{3 t}+{\mathrm e}^{7 t}-6 \,{\mathrm e}^{3 t} t \\ y \left (t \right ) &= -{\mathrm e}^{3 t}+{\mathrm e}^{7 t}+2 \,{\mathrm e}^{3 t} t \\ \end{align*} |
Problem 14035
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )\\ y^{\prime }\left (t \right )&=4 x \left (t \right )+24 t \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 3 \sin \left (2 t \right )-6 t \\ y \left (t \right ) &= -6 \cos \left (2 t \right )+6 \\ \end{align*} |
Problem 14036
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-13 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+152 \cos \left (t \right )^{4}-152 \cos \left (t \right )^{2}+19-104 \cos \left (t \right )^{3} \sin \left (t \right )+52 \cos \left (t \right ) \sin \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 13, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= 13 \sin \left (4 t \right )+13 \cos \left (4 t \right ) \\ y \left (t \right ) &= 8 \sin \left (4 t \right ) \\ \end{align*} |
Problem 14037
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 x \left (t \right )+3 y \left (t \right )+5 \operatorname {Heaviside}\left (-2+t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+6 y \left (t \right )+17 \operatorname {Heaviside}\left (-2+t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = 0, y \left (0\right ) = 0] \] |
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \operatorname {Heaviside}\left (t -2\right )+2 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{7 t -14}-3 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{3 t -6} \\ y \left (t \right ) &= -3 \operatorname {Heaviside}\left (t -2\right )+2 \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{7 t -14}+\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{3 t -6} \\ \end{align*} |
Problem 14038
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )+4 y \left (t \right )\\ y^{\prime }\left (t \right )&=8 x \left (t \right )+y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{9 t} \\ y \left (t \right ) &= -2 c_{1} {\mathrm e}^{-3 t}+c_{2} {\mathrm e}^{9 t} \\ \end{align*} |
Problem 14039
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-7 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}-\frac {c_{1} {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {9 c_{2} {\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}}}{10}+\frac {9 c_{1} {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}}}{10} \\ \end{align*} |
Problem 14040
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-5 y \left (t \right )+4\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-7 y \left (t \right )+5 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}} c_{2} +{\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} c_{1} +3 \\ y \left (t \right ) &= \frac {{\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} c_{1} \sqrt {21}}{10}-\frac {{\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}} c_{2} \sqrt {21}}{10}+\frac {9 \,{\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} c_{1}}{10}+\frac {9 \,{\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}} c_{2}}{10}+2 \\ \end{align*} |
Problem 14041
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=6 x \left (t \right )+2 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} +c_{2} {\mathrm e}^{5 t} \\ y \left (t \right ) &= 2 c_{2} {\mathrm e}^{5 t}-3 c_{1} \\ \end{align*} |
Problem 14042
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right ) y \left (t \right )-6 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right )-5 \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 14043
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-3 t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}-c_{2} {\mathrm e}^{-3 t} \\ \end{align*} |
Problem 14044
| ODE |
\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=x^{3}} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{x}}{3}-\frac {x^{3}}{2}-\frac {3 x^{2}}{4}-\frac {9 x}{4}-\frac {15}{8} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {{\mathrm e}^{-2 x} \left (\left (x^{3}+\frac {3}{2} x^{2}+\frac {9}{2} x +\frac {15}{4}\right ) {\mathrm e}^{2 x}-2 c_{2} {\mathrm e}^{3 x}-2 c_{1} \right )}{2} \] |
Problem 14045
| ODE |
\[ \boxed {y y^{\prime }+y^{4}=\sin \left (x \right )} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {\left (c_{1} \operatorname {MathieuC}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+\operatorname {MathieuS}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )\right ) \left (c_{1} \operatorname {MathieuCPrime}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+\operatorname {MathieuSPrime}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )\right )}}{2 c_{1} \operatorname {MathieuC}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+2 \operatorname {MathieuS}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )} \\ y \left (x \right ) &= \frac {\sqrt {\left (c_{1} \operatorname {MathieuC}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+\operatorname {MathieuS}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )\right ) \left (c_{1} \operatorname {MathieuCPrime}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+\operatorname {MathieuSPrime}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )\right )}}{2 c_{1} \operatorname {MathieuC}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )+2 \operatorname {MathieuS}\left (0, 8, -\frac {\pi }{4}+\frac {x}{2}\right )} \\ \end{align*} |
Problem 14046
| ODE |
\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }+5 y^{\prime }+y={\mathrm e}^{x}} \] |
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program solution |
\[ y = {\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}{6}+\frac {22}{3 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}{12}-\frac {11}{3 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}{6}-\frac {22}{3 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} +\frac {{\mathrm e}^{x}}{5} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{x}}{5}+c_{1} {\mathrm e}^{\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-101 \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-484 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}+1936\right ) x}{2904}}+c_{2} {\mathrm e}^{-\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-101 \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-484 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}-3872\right ) x}{5808}} \cos \left (\frac {\left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}} \sqrt {3}\, \left (15 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {23}-101 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+484\right ) x}{5808}\right )+c_{3} {\mathrm e}^{-\frac {\left (15 \sqrt {69}\, \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-101 \left (404+60 \sqrt {69}\right )^{\frac {2}{3}}-484 \left (404+60 \sqrt {69}\right )^{\frac {1}{3}}-3872\right ) x}{5808}} \sin \left (\frac {\left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}} \sqrt {3}\, \left (15 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}} \sqrt {3}\, \sqrt {23}-101 \left (404+60 \sqrt {3}\, \sqrt {23}\right )^{\frac {1}{3}}+484\right ) x}{5808}\right ) \] |
Problem 14047
| ODE |
\[ \boxed {{y^{\prime }}^{2}+y=0} \] |
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program solution |
\[ -2 \sqrt {-y} = x +c_{1} \] Verified OK. \[ 2 \sqrt {-y} = x +c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\left (x -c_{1} \right )^{2}}{4} \\ \end{align*} |
Problem 14048
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }+t y^{\prime }+2 y=0} \] |
|
program solution |
\[ y = t^{-i \sqrt {2}} c_{1} -\frac {i c_{2} \sqrt {2}\, t^{i \sqrt {2}}}{4} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = c_{1} \sin \left (\sqrt {2}\, \ln \left (t \right )\right )+c_{2} \cos \left (\sqrt {2}\, \ln \left (t \right )\right ) \] |
Problem 14049
| ODE |
\[ \boxed {x {y^{\prime \prime }}^{2}+2 y=2 x} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 14050
| ODE |
\[ \boxed {x^{\prime \prime }+2 \sin \left (x\right )=\sin \left (2 t \right )} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 14051
| ODE |
\[ \boxed {-y^{\prime }=1-2 x} \] |
|
program solution |
\[ y = \left (x -1\right ) x +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{2}+c_{1} -x \] |
Problem 14052
| ODE |
\[ \boxed {-y-y y^{\prime }=-2 x} \] |
|
program solution |
\[ \frac {\ln \left (y-x \right )}{3}+\frac {2 \ln \left (y+2 x \right )}{3} = c_{1} \] Verified OK.
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|
Maple solution | \begin{align*} y \left (x \right ) &= x \left (-1+\frac {\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}}{2 c_{1} x}+\frac {2 c_{1} x}{\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}}\right ) \\ y \left (x \right ) &= \frac {4 c_{1}^{2} x^{2}-2 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}+\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{2 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 c_{1}^{2} x^{2}-2 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}+\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{2 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}} \sqrt {3}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}-4 c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}\right ) x}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}} \sqrt {3}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}-4 c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}\right ) x}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}} \sqrt {3}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {2}{3}}-4 c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}}\right ) x}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{\frac {1}{3}} c_{1}} \\ \end{align*} |
Problem 14053
| ODE |
\[ \boxed {y^{\prime }+2 y=0} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{-2 x}}{c_{1}^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-2 x} c_{1} \] |
Problem 14054
| 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 14055
| ODE |
\[ \boxed {y^{\prime }+y=\sin \left (x \right )} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{-x} \left ({\mathrm e}^{x} \sin \left (x \right )-{\mathrm e}^{x} \cos \left (x \right )+2 c_{1} \right )}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2}+c_{1} {\mathrm e}^{-x} \] |
Problem 14056
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime }-12 y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{-3 t}+\frac {c_{2} {\mathrm e}^{4 t}}{7} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \left ({\mathrm e}^{7 t} c_{2} +c_{1} \right ) {\mathrm e}^{-3 t} \] |
Problem 14057
| ODE |
\[ \boxed {y^{\prime \prime }+9 y^{\prime }=0} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{-9 t}}{c_{2}^{9}}+\frac {c_{1}}{9} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = c_{1} +c_{2} {\mathrm e}^{-9 t} \] |
Problem 14058
| ODE |
\[ \boxed {x^{\prime \prime }+2 x^{\prime }-10 x=0} \] |
|
program solution |
\[ x = c_{1} {\mathrm e}^{-\left (1+\sqrt {11}\right ) t}+\frac {c_{2} \sqrt {11}\, {\mathrm e}^{\left (-1+\sqrt {11}\right ) t}}{22} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = c_{1} {\mathrm e}^{\left (-1+\sqrt {11}\right ) t}+c_{2} {\mathrm e}^{-\left (1+\sqrt {11}\right ) t} \] |
Problem 14059
| ODE |
\[ \boxed {x^{\prime \prime }+x=t \cos \left (t \right )-\cos \left (t \right )} \] |
|
program solution |
\[ x = c_{1} \cos \left (t \right )+c_{2} \sin \left (t \right )+\frac {t \cos \left (t \right )}{4}-\frac {t \sin \left (t \right )}{2}+\frac {t^{2} \sin \left (t \right )}{4} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {\left (t^{2}+4 c_{2} -2 t -1\right ) \sin \left (t \right )}{4}+\frac {\cos \left (t \right ) \left (t +4 c_{1} -2\right )}{4} \] |
Problem 14060
| ODE |
\[ \boxed {y^{\prime \prime }-12 y^{\prime }+40 y=0} \] |
|
program solution |
\[ y = c_{1} {\mathrm e}^{6 x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{6 x} \sin \left (2 x \right )}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = {\mathrm e}^{6 x} \left (c_{1} \sin \left (2 x \right )+c_{2} \cos \left (2 x \right )\right ) \] |
Problem 14061
| ODE |
\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime \prime }=0} \] |
|
program solution |
\[ y = c_{2} x +c_{1} +{\mathrm e}^{4 x} c_{3} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} +c_{2} x +c_{3} {\mathrm e}^{4 x} \] |
Problem 14062
| ODE |
\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }=0} \] |
|
program solution |
\[ y = c_{2} x +c_{1} +{\mathrm e}^{2 x} c_{3} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = c_{1} +c_{2} x +c_{3} {\mathrm e}^{2 x} \] |
Problem 14063
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-12 y^{\prime } x +42 y=0} \] |
|
program solution |
\[ y = c_{2} x^{7}+c_{1} x^{6} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = x^{6} \left (c_{2} x +c_{1} \right ) \] |
Problem 14064
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }+3 t y^{\prime }+5 y=0} \] |
|
program solution |
\[ y = c_{1} t^{-1-2 i}-\frac {i c_{2} t^{-1+2 i}}{4} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {c_{1} \sin \left (2 \ln \left (t \right )\right )+c_{2} \cos \left (2 \ln \left (t \right )\right )}{t} \] |
Problem 14065
| ODE |
\[ \boxed {y^{\prime }+\frac {x}{y}=0} \] |
|
program solution |
\[ -\frac {x^{2}}{2}-\frac {y^{2}}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {-x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+c_{1}} \\ \end{align*} |
Problem 14066
| ODE |
\[ \boxed {3 y \left (t^{2}+y\right )+t \left (t^{2}+6 y\right ) y^{\prime }=0} \] |
|
program solution |
\[ y \left (t^{2}+3 y\right ) t = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \frac {-t^{3}+\sqrt {t \left (t^{5}+12 c_{1} \right )}}{6 t} \\ y \left (t \right ) &= \frac {-t^{3}-\sqrt {t \left (t^{5}+12 c_{1} \right )}}{6 t} \\ \end{align*} |
Problem 14067
| ODE |
\[ \boxed {y^{\prime }+\frac {2 y}{x}=-3} \] |
|
program solution |
\[ y = \frac {-x^{3}+c_{1}}{x^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -x +\frac {c_{1}}{x^{2}} \] |
Problem 14068
| ODE |
\[ \boxed {y \cos \left (t \right )+\left (2 y+\sin \left (t \right )\right ) y^{\prime }=0} \] |
|
program solution |
\[ \sin \left (t \right ) y+y^{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= -\frac {\sin \left (t \right )}{2}-\frac {\sqrt {\sin \left (t \right )^{2}-4 c_{1}}}{2} \\ y \left (t \right ) &= -\frac {\sin \left (t \right )}{2}+\frac {\sqrt {\sin \left (t \right )^{2}-4 c_{1}}}{2} \\ \end{align*} |
Problem 14069
| ODE |
\[ \boxed {\frac {y}{x}+\cos \left (y\right )+\left (\ln \left (x \right )-x \sin \left (y\right )\right ) y^{\prime }=0} \] |
|
program solution |
\[ \ln \left (x \right ) y+x \cos \left (y\right ) = c_{1} \] Verified OK.
|
|
Maple solution | \[ \cos \left (y \left (x \right )\right ) x +y \left (x \right ) \ln \left (x \right )+c_{1} = 0 \] |
Problem 14070
| ODE |
\[ \boxed {y^{\prime }=\left (x^{2}-1\right ) \left (x^{3}-3 x \right )^{3}} \] |
|
program solution |
\[ y = \frac {x^{4} \left (x^{8}-12 x^{6}+54 x^{4}-108 x^{2}+81\right )}{12}+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {1}{12} x^{12}-x^{10}+\frac {9}{2} x^{8}-9 x^{6}+\frac {27}{4} x^{4}+c_{1} \] |
Problem 14071
| ODE |
\[ \boxed {y^{\prime }=\sin \left (x^{2}\right ) x} \] |
|
program solution |
\[ y = -\frac {\cos \left (x^{2}\right )}{2}+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {\cos \left (x^{2}\right )}{2}+c_{1} \] |
Problem 14072
| ODE |
\[ \boxed {y^{\prime }=\frac {x}{\sqrt {x^{2}-16}}} \] |
|
program solution |
\[ y = \sqrt {x^{2}-16}+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\left (x -4\right ) \left (x +4\right )}{\sqrt {x^{2}-16}}+c_{1} \] |
Problem 14073
| ODE |
\[ \boxed {y^{\prime }=\frac {1}{x \ln \left (x \right )}} \] |
|
program solution |
\[ y = \ln \left (\ln \left (x \right )\right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \ln \left (\ln \left (x \right )\right )+c_{1} \] |
Problem 14074
| ODE |
\[ \boxed {y^{\prime }=x \ln \left (x \right )} \] |
|
program solution |
\[ y = \frac {\ln \left (x \right ) x^{2}}{2}-\frac {x^{2}}{4}+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\ln \left (x \right ) x^{2}}{2}-\frac {x^{2}}{4}+c_{1} \] |
Problem 14075
| ODE |
\[ \boxed {y^{\prime }=x \,{\mathrm e}^{-x}} \] |
|
program solution |
\[ y = -\left (x +1\right ) {\mathrm e}^{-x}+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \left (-x -1\right ) {\mathrm e}^{-x}+c_{1} \] |
Problem 14076
| ODE |
\[ \boxed {y^{\prime }=\frac {-2 x -10}{\left (x +2\right ) \left (x -4\right )}} \] |
|
program solution |
\[ y = \ln \left (x +2\right )-3 \ln \left (x -4\right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -3 \ln \left (x -4\right )+\ln \left (x +2\right )+c_{1} \] |
Problem 14077
| ODE |
\[ \boxed {y^{\prime }=\frac {-x^{2}+x}{\left (x +1\right ) \left (x^{2}+1\right )}} \] |
|
program solution |
\[ y = \arctan \left (x \right )-\ln \left (x +1\right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\ln \left (1+x \right )+\arctan \left (x \right )+c_{1} \] |
Problem 14078
| ODE |
\[ \boxed {y^{\prime }=\frac {\sqrt {x^{2}-16}}{x}} \] |
|
program solution |
\[ y = \sqrt {x^{2}-16}-4 \arctan \left (\frac {\sqrt {x^{2}-16}}{4}\right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \sqrt {x^{2}-16}+4 \arctan \left (\frac {4}{\sqrt {x^{2}-16}}\right )+c_{1} \] |
Problem 14079
| ODE |
\[ \boxed {y^{\prime }=\left (-x^{2}+4\right )^{\frac {3}{2}}} \] |
|
program solution |
\[ y = \frac {x \left (x^{2}-10\right ) \left (x^{2}-4\right )}{4 \sqrt {-x^{2}+4}}+6 \arcsin \left (\frac {x}{2}\right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\left (-x^{3}+10 x \right ) \sqrt {-x^{2}+4}}{4}+c_{1} +6 \arcsin \left (\frac {x}{2}\right ) \] |
Problem 14080
| ODE |
\[ \boxed {y^{\prime }=\frac {1}{x^{2}-16}} \] |
|
program solution |
\[ y = -\frac {\operatorname {arctanh}\left (\frac {x}{4}\right )}{4}+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {\ln \left (x +4\right )}{8}+\frac {\ln \left (x -4\right )}{8}+c_{1} \] |
Problem 14081
| ODE |
\[ \boxed {y^{\prime }=\cos \left (x \right ) \cot \left (x \right )} \] |
|
program solution |
\[ y = \cos \left (x \right )+\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \cos \left (x \right )+\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \] |
Problem 14082
| ODE |
\[ \boxed {y^{\prime }=\sin \left (x \right )^{3} \tan \left (x \right )} \] |
|
program solution |
\[ y = -\frac {\sin \left (x \right )^{3}}{3}-\sin \left (x \right )+\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {\sin \left (x \right )^{3}}{3}-\sin \left (x \right )+\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+c_{1} \] |
Problem 14083
| ODE |
\[ \boxed {y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
|
|
program solution |
\[ -\frac {\ln \left (y\right )}{2} = x -\frac {\ln \left (2\right )}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = 2 \,{\mathrm e}^{-2 x} \] |
Problem 14084
| ODE |
\[ \boxed {y^{\prime }+y=\sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
|
|
program solution |
\[ y = \frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2}-\frac {{\mathrm e}^{-t}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {\cos \left (t \right )}{2}+\frac {\sin \left (t \right )}{2}-\frac {{\mathrm e}^{-t}}{2} \] |
Problem 14085
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime }-12 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
|
|
program solution |
\[ y = \frac {2 \,{\mathrm e}^{4 x}}{7}+\frac {5 \,{\mathrm e}^{-3 x}}{7} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\left (2 \,{\mathrm e}^{7 x}+5\right ) {\mathrm e}^{-3 x}}{7} \] |
Problem 14086
| ODE |
\[ \boxed {y^{\prime \prime }+9 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*}
|
|
program solution |
N/A |
|
Maple solution | \[ y \left (x \right ) = \frac {17}{9}+\frac {{\mathrm e}^{-9 x}}{9} \] |
Problem 14087
| ODE |
\[ \boxed {y^{\prime \prime \prime }-2 y^{\prime \prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = 3] \end {align*}
|
|
program solution |
\[ y = -\frac {3}{4}-\frac {x}{2}+\frac {3 \,{\mathrm e}^{2 x}}{4} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\frac {3}{4}-\frac {x}{2}+\frac {3 \,{\mathrm e}^{2 x}}{4} \] |
Problem 14088
| ODE |
\[ \boxed {y^{\prime \prime \prime }-4 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1, y^{\prime \prime }\left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = 1+\frac {{\mathrm e}^{-2 x}}{4}-\frac {{\mathrm e}^{2 x}}{4} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = 1-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{-2 x}}{4} \] |
Problem 14089
| ODE |
\[ \boxed {t^{2} y^{\prime \prime }-12 t y^{\prime }+42 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = -1] \end {align*}
|
|
program solution |
\[ y = -t^{7}+t^{6} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -t^{7}+t^{6} \] |
Problem 14090
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x +5 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 1] \end {align*}
|
|
program solution |
\[ y = -\frac {i \left (-x^{-1-2 i}+x^{-1+2 i}\right )}{4} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\sin \left (2 \ln \left (x \right )\right )}{2 x} \] |
Problem 14091
| ODE |
\[ \boxed {y^{\prime }=4 x^{3}-x +2} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ y = x^{4}-\frac {1}{2} x^{2}+2 x +1 \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = x^{4}-\frac {1}{2} x^{2}+1+2 x \] |
Problem 14092
| ODE |
\[ \boxed {y^{\prime }=\sin \left (2 t \right )-\cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = -\frac {\sin \left (2 t \right )}{2}-\frac {\cos \left (2 t \right )}{2}+\frac {1}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {\sin \left (2 t \right )}{2}-\frac {\cos \left (2 t \right )}{2}+\frac {1}{2} \] |
Problem 14093
| ODE |
\[ \boxed {y^{\prime }=\frac {\cos \left (\frac {1}{x}\right )}{x^{2}}} \] With initial conditions \begin {align*} \left [y \left (\frac {2}{\pi }\right ) = 1\right ] \end {align*}
|
|
program solution |
\[ y = -\sin \left (\frac {1}{x}\right )+2 \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = -\sin \left (\frac {1}{x}\right )+2 \] |
Problem 14094
| ODE |
\[ \boxed {y^{\prime }=\frac {\ln \left (x \right )}{x}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
|
|
program solution |
\[ y = \frac {\ln \left (x \right )^{2}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \frac {\ln \left (x \right )^{2}}{2} \] |
Problem 14095
| ODE |
\[ \boxed {y^{\prime }-\frac {\left (x -4\right ) y^{3}}{x^{3} \left (y-2\right )}=0} \] |
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program solution |
\[ \frac {x -2}{x^{2}}-\frac {1}{y}+\frac {1}{y^{2}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\left (x +\sqrt {8+\left (4 c_{1} +1\right ) x^{2}-4 x}\right ) x}{2 c_{1} x^{2}-2 x +4} \\ y \left (x \right ) &= \frac {x \left (-x +\sqrt {8+\left (4 c_{1} +1\right ) x^{2}-4 x}\right )}{2 c_{1} x^{2}-2 x +4} \\ \end{align*} |
Problem 14096
| ODE |
\[ \boxed {y^{\prime }-\frac {y^{2}+2 y x}{x^{2}}=0} \] |
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program solution |
\[ y = -\frac {c_{3} x^{2}}{c_{3} x +1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{2}}{c_{1} -x} \] |
Problem 14097
| ODE |
\[ \boxed {y^{\prime } x +y=\cos \left (x \right )} \] |
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program solution |
\[ y = \frac {\sin \left (x \right )+c_{1}}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sin \left (x \right )+c_{1}}{x} \] |
Problem 14098
| ODE |
\[ \boxed {16 y^{\prime \prime }+24 y^{\prime }+153 y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {3 x}{4}} \cos \left (3 x \right )+\frac {c_{2} {\mathrm e}^{-\frac {3 x}{4}} \sin \left (3 x \right )}{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {3 x}{4}} \left (c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right )\right ) \] |
Problem 14099
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ODE |
\begin {align*} x^{\prime }\left (t \right )&=4 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )-2 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (\sin \left (\sqrt {3}\, t \right ) c_{1} +\cos \left (\sqrt {3}\, t \right ) c_{2} \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-t} \left (\sqrt {3}\, \sin \left (\sqrt {3}\, t \right ) c_{2} -\sqrt {3}\, \cos \left (\sqrt {3}\, t \right ) c_{1} +\sin \left (\sqrt {3}\, t \right ) c_{1} +\cos \left (\sqrt {3}\, t \right ) c_{2} \right )}{4} \\ \end{align*} |
Problem 14100
| ODE |
\[ \boxed {4 x \left (y^{2}+x^{2}\right )-5 y+4 y \left (x^{2}+y^{2}-5 x \right ) y^{\prime }=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |