2.17.67 Problems 6601 to 6700

Problem 6601

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-y^{\prime } x +y \left (x^{2}+1\right )=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{10}+\frac {x^{4}}{360}-\frac {x^{6}}{28080}+O\left (x^{8}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{168}-\frac {x^{6}}{11088}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x^{2}+\frac {1}{168} x^{4}-\frac {1}{11088} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {1}{10} x^{2}+\frac {1}{360} x^{4}-\frac {1}{28080} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6602

ODE	\[ \boxed {3 x y^{\prime \prime }+\left (-x +2\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {x}{3}+\frac {x^{2}}{18}+\frac {x^{3}}{162}+\frac {x^{4}}{1944}+\frac {x^{5}}{29160}+\frac {x^{6}}{524880}+\frac {x^{7}}{11022480}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{10}+\frac {x^{3}}{80}+\frac {x^{4}}{880}+\frac {x^{5}}{12320}+\frac {x^{6}}{209440}+\frac {x^{7}}{4188800}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1+\frac {1}{3} x +\frac {1}{18} x^{2}+\frac {1}{162} x^{3}+\frac {1}{1944} x^{4}+\frac {1}{29160} x^{5}+\frac {1}{524880} x^{6}+\frac {1}{11022480} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {1}{2} x +\frac {1}{10} x^{2}+\frac {1}{80} x^{3}+\frac {1}{880} x^{4}+\frac {1}{12320} x^{5}+\frac {1}{209440} x^{6}+\frac {1}{4188800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6603

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (x -\frac {2}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1+\frac {3 x}{4}+\frac {9 x^{2}}{56}+\frac {9 x^{3}}{560}+\frac {27 x^{4}}{29120}+\frac {81 x^{5}}{2329600}+\frac {81 x^{6}}{88524800}+\frac {243 x^{7}}{13632819200}+O\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1+\frac {3 x}{2}+\frac {9 x^{2}}{20}+\frac {9 x^{3}}{160}+\frac {27 x^{4}}{7040}+\frac {81 x^{5}}{492800}+\frac {81 x^{6}}{16755200}+\frac {243 x^{7}}{2345728000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1+\frac {3}{2} x +\frac {9}{20} x^{2}+\frac {9}{160} x^{3}+\frac {27}{7040} x^{4}+\frac {81}{492800} x^{5}+\frac {81}{16755200} x^{6}+\frac {243}{2345728000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {2}{3}} \left (1+\frac {3}{4} x +\frac {9}{56} x^{2}+\frac {9}{560} x^{3}+\frac {27}{29120} x^{4}+\frac {81}{2329600} x^{5}+\frac {81}{88524800} x^{6}+\frac {243}{13632819200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6604

ODE	\[ \boxed {2 x y^{\prime \prime }-\left (2 x +3\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{2}} \left (1+\frac {4 x}{7}+\frac {4 x^{2}}{21}+\frac {32 x^{3}}{693}+\frac {80 x^{4}}{9009}+\frac {64 x^{5}}{45045}+\frac {64 x^{6}}{328185}+\frac {1024 x^{7}}{43648605}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {x}{3}-\frac {x^{2}}{6}-\frac {x^{3}}{6}-\frac {5 x^{4}}{72}-\frac {7 x^{5}}{360}-\frac {x^{6}}{240}-\frac {11 x^{7}}{15120}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {5}{2}} \left (1+\frac {4}{7} x +\frac {4}{21} x^{2}+\frac {32}{693} x^{3}+\frac {80}{9009} x^{4}+\frac {64}{45045} x^{5}+\frac {64}{328185} x^{6}+\frac {1024}{43648605} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+\frac {1}{3} x -\frac {1}{6} x^{2}-\frac {1}{6} x^{3}-\frac {5}{72} x^{4}-\frac {7}{360} x^{5}-\frac {1}{240} x^{6}-\frac {11}{15120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6605

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {4}{9}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1-\frac {3 x^{2}}{20}+\frac {9 x^{4}}{1280}-\frac {9 x^{6}}{56320}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {3 x^{2}}{4}+\frac {9 x^{4}}{128}-\frac {9 x^{6}}{3584}+O\left (x^{8}\right )\right )}{x^{\frac {2}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {3}{20} x^{2}+\frac {9}{1280} x^{4}-\frac {9}{56320} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1-\frac {3}{4} x^{2}+\frac {9}{128} x^{4}-\frac {9}{3584} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{\frac {2}{3}}} \]

Problem 6606

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+9 x^{2} y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1-\frac {x}{2}+\frac {5 x^{2}}{28}-\frac {x^{3}}{21}+\frac {11 x^{4}}{1092}-\frac {11 x^{5}}{6240}+\frac {187 x^{6}}{711360}-\frac {17 x^{7}}{497952}+O\left (x^{8}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {x}{2}+\frac {x^{2}}{5}-\frac {7 x^{3}}{120}+\frac {7 x^{4}}{528}-\frac {13 x^{5}}{5280}+\frac {13 x^{6}}{33660}-\frac {247 x^{7}}{4712400}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {1}{2} x +\frac {1}{5} x^{2}-\frac {7}{120} x^{3}+\frac {7}{528} x^{4}-\frac {13}{5280} x^{5}+\frac {13}{33660} x^{6}-\frac {247}{4712400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {2}{3}} \left (1-\frac {1}{2} x +\frac {5}{28} x^{2}-\frac {1}{21} x^{3}+\frac {11}{1092} x^{4}-\frac {11}{6240} x^{5}+\frac {187}{711360} x^{6}-\frac {17}{497952} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6607

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+3 y^{\prime } x +\left (2 x -1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {2 x}{5}+\frac {2 x^{2}}{35}-\frac {4 x^{3}}{945}+\frac {2 x^{4}}{10395}-\frac {4 x^{5}}{675675}+\frac {4 x^{6}}{30405375}-\frac {8 x^{7}}{3618239625}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+2 x -2 x^{2}+\frac {4 x^{3}}{9}-\frac {2 x^{4}}{45}+\frac {4 x^{5}}{1575}-\frac {4 x^{6}}{42525}+\frac {8 x^{7}}{3274425}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{2}} \left (1-\frac {2}{5} x +\frac {2}{35} x^{2}-\frac {4}{945} x^{3}+\frac {2}{10395} x^{4}-\frac {4}{675675} x^{5}+\frac {4}{30405375} x^{6}-\frac {8}{3618239625} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1+2 x -2 x^{2}+\frac {4}{9} x^{3}-\frac {2}{45} x^{4}+\frac {4}{1575} x^{5}-\frac {4}{42525} x^{6}+\frac {8}{3274425} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6608

ODE	\[ \boxed {x y^{\prime \prime }+2 y^{\prime }-y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{120}+\frac {x^{6}}{5040}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{24}+\frac {x^{6}}{720}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 6609

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-\frac {1}{4}+x^{2}\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{120}-\frac {x^{6}}{5040}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{24}-\frac {x^{6}}{720}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}-\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

Problem 6610

ODE	\[ \boxed {x y^{\prime \prime }-y^{\prime } x +y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+O\left (x^{8}\right )\right )+c_{2} \left (-x \left (1+O\left (x^{8}\right )\right ) \ln \left (x \right )+1-\frac {x^{2}}{2}-\frac {x^{3}}{12}-\frac {x^{4}}{72}-\frac {x^{5}}{480}-\frac {x^{6}}{3600}-\frac {x^{7}}{30240}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1+\operatorname {O}\left (x^{8}\right )\right )+\left (-x +\operatorname {O}\left (x^{8}\right )\right ) \ln \left (x \right ) c_{2} +\left (1+x -\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {1}{72} x^{4}-\frac {1}{480} x^{5}-\frac {1}{3600} x^{6}-\frac {1}{30240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6611

ODE	\[ \boxed {y^{\prime \prime }+\frac {3 y^{\prime }}{x}-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{4}+\frac {x^{4}}{48}+\frac {x^{6}}{1152}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1+\frac {x^{2}}{4}+\frac {x^{4}}{48}+\frac {x^{6}}{1152}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {1-\frac {3 x^{4}}{16}-\frac {7 x^{6}}{288}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{4} x^{2}+\frac {1}{48} x^{4}+\frac {1}{1152} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (\left (-2\right ) x^{2}-\frac {1}{2} x^{4}-\frac {1}{24} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2+\frac {3}{8} x^{4}+\frac {7}{144} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

Problem 6612

ODE	\[ \boxed {x y^{\prime \prime }+\left (1-x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \ln \left (x \right )-x -\frac {3 x^{2}}{4}-\frac {11 x^{3}}{36}-\frac {25 x^{4}}{288}-\frac {137 x^{5}}{7200}-\frac {49 x^{6}}{14400}-\frac {121 x^{7}}{235200}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-x -\frac {3}{4} x^{2}-\frac {11}{36} x^{3}-\frac {25}{288} x^{4}-\frac {137}{7200} x^{5}-\frac {49}{14400} x^{6}-\frac {121}{235200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6613

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-x +\frac {x^{2}}{4}-\frac {x^{3}}{36}+\frac {x^{4}}{576}-\frac {x^{5}}{14400}+\frac {x^{6}}{518400}-\frac {x^{7}}{25401600}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1-x +\frac {x^{2}}{4}-\frac {x^{3}}{36}+\frac {x^{4}}{576}-\frac {x^{5}}{14400}+\frac {x^{6}}{518400}-\frac {x^{7}}{25401600}+O\left (x^{8}\right )\right ) \ln \left (x \right )+2 x -\frac {3 x^{2}}{4}+\frac {11 x^{3}}{108}-\frac {25 x^{4}}{3456}+\frac {137 x^{5}}{432000}-\frac {49 x^{6}}{5184000}+\frac {121 x^{7}}{592704000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {1}{36} x^{3}+\frac {1}{576} x^{4}-\frac {1}{14400} x^{5}+\frac {1}{518400} x^{6}-\frac {1}{25401600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {11}{108} x^{3}-\frac {25}{3456} x^{4}+\frac {137}{432000} x^{5}-\frac {49}{5184000} x^{6}+\frac {121}{592704000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 6614

ODE	\[ \boxed {x y^{\prime \prime }+\left (x -6\right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{7} \left (1-\frac {x}{2}+\frac {5 x^{2}}{36}-\frac {x^{3}}{36}+\frac {7 x^{4}}{1584}-\frac {7 x^{5}}{11880}+\frac {7 x^{6}}{102960}-\frac {x^{7}}{144144}+O\left (x^{8}\right )\right )+c_{2} \left (1-\frac {x}{2}+\frac {x^{2}}{10}-\frac {x^{3}}{120}+\frac {x^{7}}{100800}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{7} \left (1-\frac {1}{2} x +\frac {5}{36} x^{2}-\frac {1}{36} x^{3}+\frac {7}{1584} x^{4}-\frac {7}{11880} x^{5}+\frac {7}{102960} x^{6}-\frac {1}{144144} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (3628800-1814400 x +362880 x^{2}-30240 x^{3}+36 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6615

ODE	\[ \boxed {\left (x -1\right ) x y^{\prime \prime }+3 y^{\prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{4} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+7 x^{6}+8 x^{7}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {2 x}{3}+\frac {x^{2}}{3}-\frac {x^{4}}{3}-\frac {2 x^{5}}{3}-x^{6}-\frac {4 x^{7}}{3}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{4} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+7 x^{6}+8 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-144-96 x -48 x^{2}+48 x^{4}+96 x^{5}+144 x^{6}+192 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6616

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 y^{\prime }}{t}+\lambda y=0} \] With the expansion point for the power series method at \(t = 0\).
program solution	\[ y = c_{1} \left (1-\frac {\lambda \,t^{2}}{6}+\frac {\lambda ^{2} t^{4}}{120}-\frac {\lambda ^{3} t^{6}}{5040}+O\left (t^{8}\right )\right )+\frac {c_{2} \left (1-\frac {\lambda \,t^{2}}{2}+\frac {\lambda ^{2} t^{4}}{24}-\frac {\lambda ^{3} t^{6}}{720}+O\left (t^{8}\right )\right )}{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \left (1-\frac {1}{6} \lambda \,t^{2}+\frac {1}{120} \lambda ^{2} t^{4}-\frac {1}{5040} \lambda ^{3} t^{6}+\operatorname {O}\left (t^{8}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} \lambda \,t^{2}+\frac {1}{24} \lambda ^{2} t^{4}-\frac {1}{720} \lambda ^{3} t^{6}+\operatorname {O}\left (t^{8}\right )\right )}{t} \]

Problem 6617

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

Problem 6618

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

Problem 6619

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

Problem 6620

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

Problem 6621

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

Problem 6622

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

Problem 6623

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

Problem 6624

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

Problem 6625

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

Problem 6626

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (36 x^{2}-\frac {1}{4}\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \cos \left (6 x \right )}{\sqrt {x}}+\frac {c_{2} \sin \left (6 x \right )}{6 \sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (6 x \right )+c_{2} \cos \left (6 x \right )}{\sqrt {x}} \]

Problem 6627

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (25 x^{2}-\frac {4}{9}\right ) y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (-\frac {2}{3}, 5 x \right )+c_{2} \operatorname {BesselY}\left (-\frac {2}{3}, 5 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (\frac {2}{3}, 5 x \right )+c_{2} \operatorname {BesselY}\left (\frac {2}{3}, 5 x \right ) \]

Problem 6628

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (2 x^{2}-64\right ) y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (8, \sqrt {2}\, x \right )+c_{2} \operatorname {BesselY}\left (8, \sqrt {2}\, x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-16 \sqrt {2}\, c_{1} \left (x^{6}-75 x^{4}+1080 x^{2}-2520\right ) \operatorname {BesselJ}\left (1, \sqrt {2}\, x \right )-16 \sqrt {2}\, c_{2} \left (x^{6}-75 x^{4}+1080 x^{2}-2520\right ) \operatorname {BesselY}\left (1, \sqrt {2}\, x \right )+x \left (x^{6}-240 x^{4}+7200 x^{2}-40320\right ) \left (\operatorname {BesselJ}\left (0, \sqrt {2}\, x \right ) c_{1} +\operatorname {BesselY}\left (0, \sqrt {2}\, x \right ) c_{2} \right )}{x^{7}} \]

Problem 6629

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

Problem 6630

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

Problem 6631

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

Problem 6632

ODE	\[ \boxed {x y^{\prime \prime }-5 y^{\prime }+y x=0} \]
program solution	\[ y = c_{1} x^{3} \operatorname {BesselJ}\left (3, x\right )+c_{2} x^{3} \operatorname {BesselY}\left (3, x\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\left (c_{1} \left (x^{2}-8\right ) \operatorname {BesselJ}\left (1, x\right )+c_{2} \left (x^{2}-8\right ) \operatorname {BesselY}\left (1, x\right )+4 x \left (c_{1} \operatorname {BesselJ}\left (0, x\right )+c_{2} \operatorname {BesselY}\left (0, x\right )\right )\right ) x \]

Problem 6633

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

Problem 6634

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+\left (16 x^{2}+1\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (0, 2 x \right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (0, 2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\operatorname {BesselJ}\left (0, 2 x \right ) c_{1} +\operatorname {BesselY}\left (0, 2 x \right ) c_{2} \right ) \sqrt {x} \]

Problem 6635

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

Problem 6636

ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+9 y^{\prime } x +\left (x^{6}-36\right ) y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (-\frac {2}{3}, \frac {x^{3}}{9}\right )+c_{2} \operatorname {BesselY}\left (-\frac {2}{3}, \frac {x^{3}}{9}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (\frac {2}{3}, \frac {x^{3}}{9}\right )+c_{2} \operatorname {BesselY}\left (\frac {2}{3}, \frac {x^{3}}{9}\right ) \]

Problem 6637

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

Problem 6638

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime }-7 x^{3} y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselI}\left (0, \frac {\sqrt {7}\, x^{2}}{2}\right )+c_{2} \operatorname {BesselY}\left (0, \frac {i \sqrt {7}\, x^{2}}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselI}\left (0, \frac {\sqrt {7}\, x^{2}}{2}\right )+c_{2} \operatorname {BesselK}\left (0, \frac {\sqrt {7}\, x^{2}}{2}\right ) \]

Problem 6639

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

Problem 6640

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

Problem 6641

ODE	\[ \boxed {16 x^{2} y^{\prime \prime }+32 y^{\prime } x +\left (x^{4}-12\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{-\frac {i x^{2}}{8}}}{x^{\frac {3}{2}}}-\frac {2 i c_{2} {\mathrm e}^{\frac {i x^{2}}{8}}}{x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (\frac {x^{2}}{8}\right )+c_{2} \cos \left (\frac {x^{2}}{8}\right )}{x^{\frac {3}{2}}} \]

Problem 6642

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-4 y^{\prime } x +\left (16 x^{2}+3\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \cos \left (2 x \right )+\frac {c_{2} \sqrt {x}\, \sin \left (2 x \right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} \sin \left (2 x \right )+c_{2} \cos \left (2 x \right )\right ) \]

Problem 6643

ODE	\[ \boxed {2 x y^{\prime \prime }+y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{3}+\frac {x^{2}}{30}-\frac {x^{3}}{630}+\frac {x^{4}}{22680}-\frac {x^{5}}{1247400}+\frac {x^{6}}{97297200}-\frac {x^{7}}{10216206000}+O\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {x^{2}}{6}-\frac {x^{3}}{90}+\frac {x^{4}}{2520}-\frac {x^{5}}{113400}+\frac {x^{6}}{7484400}-\frac {x^{7}}{681080400}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\frac {1}{97297200} x^{6}-\frac {1}{10216206000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\frac {1}{7484400} x^{6}-\frac {1}{681080400} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6644

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\frac {1}{384} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{15} x^{5}+\frac {1}{105} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}\right ) c_{1} +\left (x +\frac {1}{3} x^{3}+\frac {1}{15} x^{5}+\frac {1}{105} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}+\frac {1}{15} x^{5}+\frac {1}{105} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6645

ODE	\[ \boxed {\left (x -1\right ) y^{\prime \prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {3}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{8} x^{4}+\frac {9}{20} x^{5}+\frac {29}{80} x^{6}+\frac {163}{560} x^{7}+\frac {213}{896} x^{8}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {9}{40} x^{5}+\frac {7}{40} x^{6}+\frac {79}{560} x^{7}+\frac {129}{1120} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+\frac {3}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{8} x^{4}+\frac {9}{20} x^{5}+\frac {29}{80} x^{6}+\frac {163}{560} x^{7}\right ) c_{1} +\left (x +\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {9}{40} x^{5}+\frac {7}{40} x^{6}+\frac {79}{560} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3}{2} x^{2}+\frac {1}{2} x^{3}+\frac {5}{8} x^{4}+\frac {9}{20} x^{5}+\frac {29}{80} x^{6}+\frac {163}{560} x^{7}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{3}+\frac {1}{4} x^{4}+\frac {9}{40} x^{5}+\frac {7}{40} x^{6}+\frac {79}{560} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6646

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

Problem 6647

ODE	\[ \boxed {x y^{\prime \prime }-\left (x +2\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+\frac {x}{4}+\frac {x^{2}}{20}+\frac {x^{3}}{120}+\frac {x^{4}}{840}+\frac {x^{5}}{6720}+\frac {x^{6}}{60480}+\frac {x^{7}}{604800}+O\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+\frac {1}{4} x +\frac {1}{20} x^{2}+\frac {1}{120} x^{3}+\frac {1}{840} x^{4}+\frac {1}{6720} x^{5}+\frac {1}{60480} x^{6}+\frac {1}{604800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (12+12 x +6 x^{2}+2 x^{3}+\frac {1}{2} x^{4}+\frac {1}{10} x^{5}+\frac {1}{60} x^{6}+\frac {1}{420} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 6648

ODE	\[ \boxed {\cos \left (x \right ) y^{\prime \prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{720} x^{6}+\frac {13}{40320} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{60} x^{5}-\frac {13}{5040} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{720} x^{6}\right ) c_{1} +\left (x -\frac {1}{6} x^{3}-\frac {1}{60} x^{5}-\frac {13}{5040} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{720} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{60} x^{5}-\frac {13}{5040} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6649

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x^{4}+x^{3}-3 x^{2}-2 x +3-\frac {x^{5}}{4}-\frac {x^{6}}{5}+\frac {x^{7}}{24}+\frac {x^{8}}{35}+O\left (x^{8}\right ) \] Verified OK. \[ y = 3-3 x^{2}+x^{4}-\frac {x^{6}}{5}-2 x +x^{3}-\frac {x^{5}}{4}+\frac {x^{7}}{24}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3-2 x -3 x^{2}+x^{3}+x^{4}-\frac {1}{4} x^{5}-\frac {1}{5} x^{6}+\frac {1}{24} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6650

ODE	\[ \boxed {\left (x +2\right ) y^{\prime \prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x -\frac {x^{3}}{4}+\frac {x^{4}}{16}-\frac {x^{6}}{320}+\frac {x^{7}}{896}-\frac {3 x^{8}}{8960}+O\left (x^{8}\right ) \] Verified OK. \[ y = x -\frac {x^{3}}{4}+\frac {x^{4}}{16}-\frac {x^{6}}{320}+\frac {x^{7}}{896}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {1}{4} x^{3}+\frac {1}{16} x^{4}-\frac {1}{320} x^{6}+\frac {1}{896} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

Problem 6651

ODE	\[ \boxed {\left (1-2 \sin \left (x \right )\right ) y^{\prime \prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{6} x^{4}-\frac {1}{5} x^{5}-\frac {1}{4} x^{6}-\frac {85}{252} x^{7}-\frac {1609}{3360} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}-\frac {1}{10} x^{5}-\frac {2}{15} x^{6}-\frac {13}{72} x^{7}-\frac {9}{35} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}-\frac {1}{6} x^{4}-\frac {1}{5} x^{5}-\frac {1}{4} x^{6}-\frac {85}{252} x^{7}\right ) c_{1} +\left (x -\frac {1}{12} x^{4}-\frac {1}{10} x^{5}-\frac {2}{15} x^{6}-\frac {13}{72} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}-\frac {1}{6} x^{4}-\frac {1}{5} x^{5}-\frac {1}{4} x^{6}-\frac {85}{252} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}-\frac {1}{10} x^{5}-\frac {2}{15} x^{6}-\frac {13}{72} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6652

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -6, y^{\prime }\left (1\right ) = 3] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = -9+3 x +\frac {3 \left (x -1\right )^{2}}{2}-\frac {3 \left (x -1\right )^{3}}{2}+\frac {3 \left (x -1\right )^{5}}{10}-\frac {\left (x -1\right )^{6}}{20}-\frac {\left (x -1\right )^{7}}{28}+\frac {3 \left (x -1\right )^{8}}{280}+O\left (\left (x -1\right )^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -6+3 \left (x -1\right )+\frac {3}{2} \left (x -1\right )^{2}-\frac {3}{2} \left (x -1\right )^{3}+\frac {3}{10} \left (x -1\right )^{5}-\frac {1}{20} \left (x -1\right )^{6}-\frac {1}{28} \left (x -1\right )^{7}+\operatorname {O}\left (\left (x -1\right )^{8}\right ) \]

Problem 6653

ODE	\[ \boxed {x y^{\prime \prime }+\left (1-\cos \left (x \right )\right ) y^{\prime }+y x^{2}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{80} x^{5}+\frac {1}{180} x^{6}-\frac {5}{4032} x^{7}-\frac {1}{1920} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}+\frac {1}{120} x^{6}+\frac {73}{60480} x^{7}-\frac {17}{20160} x^{8}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{80} x^{5}+\frac {1}{180} x^{6}-\frac {5}{4032} x^{7}\right ) c_{1} +\left (x -\frac {1}{12} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}+\frac {1}{120} x^{6}+\frac {73}{60480} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{6} x^{3}+\frac {1}{80} x^{5}+\frac {1}{180} x^{6}-\frac {5}{4032} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{3}-\frac {1}{12} x^{4}+\frac {1}{120} x^{5}+\frac {1}{120} x^{6}+\frac {73}{60480} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 6654

ODE	\[ \boxed {\left ({\mathrm e}^{x}-1-x \right ) y^{\prime \prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-x +\frac {4 x^{2}}{9}-\frac {29 x^{3}}{216}+\frac {37 x^{4}}{1200}-\frac {58 x^{5}}{10125}+\frac {14209 x^{6}}{15876000}-\frac {107329 x^{7}}{889056000}+O\left (x^{8}\right )\right )+c_{2} \left (x \left (1-x +\frac {4 x^{2}}{9}-\frac {29 x^{3}}{216}+\frac {37 x^{4}}{1200}-\frac {58 x^{5}}{10125}+\frac {14209 x^{6}}{15876000}-\frac {107329 x^{7}}{889056000}+O\left (x^{8}\right )\right ) \ln \left (x \right )+1+\frac {4 x^{2}}{3}-\frac {175 x^{3}}{216}+\frac {3727 x^{4}}{12960}-\frac {47531 x^{5}}{648000}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (1-x +\frac {4}{9} x^{2}-\frac {29}{216} x^{3}+\frac {37}{1200} x^{4}-\frac {58}{10125} x^{5}+\frac {14209}{15876000} x^{6}-\frac {107329}{889056000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-2\right ) x +2 x^{2}-\frac {8}{9} x^{3}+\frac {29}{108} x^{4}-\frac {37}{600} x^{5}+\frac {116}{10125} x^{6}-\frac {14209}{7938000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {8}{3} x^{2}+\frac {175}{108} x^{3}-\frac {3727}{6480} x^{4}+\frac {47531}{324000} x^{5}-\frac {3003737}{102060000} x^{6}+\frac {48833381}{10001880000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 6655

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x^{2}+2 y x=10 x^{3}-2 x +5} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{18} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}+\frac {1}{28} x^{7}\right ) y^{\prime }\left (0\right )+\frac {5 x^{2}}{2}-\frac {x^{3}}{3}+\frac {x^{6}}{18}+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{18} x^{6}\right ) c_{1} +\left (x -\frac {1}{4} x^{4}+\frac {1}{28} x^{7}\right ) c_{2} +\frac {5 x^{2}}{2}-\frac {x^{3}}{3}+\frac {x^{6}}{18}+O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{3} x^{3}+\frac {1}{18} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}+\frac {1}{28} x^{7}\right ) D\left (y \right )\left (0\right )+\frac {5 x^{2}}{2}-\frac {x^{3}}{3}+\frac {x^{6}}{18}+O\left (x^{8}\right ) \]

Problem 6656

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

Problem 6657

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

Problem 6658

ODE	\[ \boxed {y^{\prime }+6 y={\mathrm e}^{4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{4 t}}{10}+\frac {19 \,{\mathrm e}^{-6 t}}{10} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left ({\mathrm e}^{10 t}+19\right ) {\mathrm e}^{-6 t}}{10} \]

Problem 6659

ODE	\[ \boxed {y^{\prime }-y=2 \cos \left (5 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{t}}{13}-\frac {\cos \left (5 t \right )}{13}+\frac {5 \sin \left (5 t \right )}{13} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{t}}{13}-\frac {\cos \left (5 t \right )}{13}+\frac {5 \sin \left (5 t \right )}{13} \]

Problem 6660

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

Problem 6661

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }=6 \,{\mathrm e}^{3 t}-3 \,{\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {11 \,{\mathrm e}^{4 t}}{10}-2 \,{\mathrm e}^{3 t}+\frac {5}{2}-\frac {3 \,{\mathrm e}^{-t}}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {5}{2}+\frac {11 \,{\mathrm e}^{4 t}}{10}-2 \,{\mathrm e}^{3 t}-\frac {3 \,{\mathrm e}^{-t}}{5} \]

Problem 6662

ODE	\[ \boxed {y^{\prime \prime }+y=\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 10, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 10 \cos \left (t \right )+2 \sin \left (t \right )-\sqrt {2}\, \sin \left (\sqrt {2}\, t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\sqrt {2}\, \sin \left (\sqrt {2}\, t \right )+10 \cos \left (t \right )+2 \sin \left (t \right ) \]

Problem 6663

ODE	\[ \boxed {y^{\prime \prime }+9 y={\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {\cos \left (3 t \right )}{10}-\frac {\sin \left (3 t \right )}{30}+\frac {{\mathrm e}^{t}}{10} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\cos \left (3 t \right )}{10}-\frac {\sin \left (3 t \right )}{30}+\frac {{\mathrm e}^{t}}{10} \]

Problem 6664

ODE	\[ \boxed {2 y^{\prime \prime \prime }+3 y^{\prime \prime }-3 y^{\prime }-2 y={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {7 \cosh \left (t \right )}{9}-\frac {2 \sinh \left (t \right )}{9}+\frac {{\mathrm e}^{-2 t}}{9}-\frac {8 \,{\mathrm e}^{-\frac {t}{2}}}{9} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {7 \cosh \left (t \right )}{9}-\frac {2 \sinh \left (t \right )}{9}+\frac {{\mathrm e}^{-2 t}}{9}-\frac {8 \,{\mathrm e}^{-\frac {t}{2}}}{9} \]

Problem 6665

ODE	\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }-y^{\prime }-2 y=\sin \left (3 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0, y^{\prime \prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {3 \cos \left (3 t \right )}{130}-\frac {\sin \left (3 t \right )}{65}-\frac {13 \cosh \left (t \right )}{30}+\frac {13 \sinh \left (t \right )}{15}+\frac {16 \,{\mathrm e}^{-2 t}}{39} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {3 \cos \left (3 t \right )}{130}-\frac {\sin \left (3 t \right )}{65}-\frac {13 \cosh \left (t \right )}{30}+\frac {13 \sinh \left (t \right )}{15}+\frac {16 \,{\mathrm e}^{-2 t}}{39} \]

Problem 6666

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

Problem 6667

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

Problem 6668

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

Problem 6669

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

Problem 6670

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

Problem 6671

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=t^{3} {\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {t^{5} {\mathrm e}^{2 t}}{20} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{5} {\mathrm e}^{2 t}}{20} \]

Problem 6672

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (30 t -2\right ) {\mathrm e}^{3 t}}{27}+\frac {t}{9}+\frac {2}{27} \]

Problem 6673

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=t^{3}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (-13 t +2\right ) {\mathrm e}^{2 t}}{8}+\frac {t^{3}}{4}+\frac {3 t^{2}}{4}+\frac {9 t}{8}+\frac {3}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (2-13 t \right ) {\mathrm e}^{2 t}}{8}+\frac {t^{3}}{4}+\frac {3 t^{2}}{4}+\frac {9 t}{8}+\frac {3}{4} \]

Problem 6674

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+13 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -3] \end {align*}
program solution	\[ y = -\frac {3 \,{\mathrm e}^{3 t} \sin \left (2 t \right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3 \,{\mathrm e}^{3 t} \sin \left (2 t \right )}{2} \]

Problem 6675

ODE	\[ \boxed {2 y^{\prime \prime }+20 y^{\prime }+51 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{-5 t} \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+5 \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 \,{\mathrm e}^{-5 t} \left (\cos \left (\frac {\sqrt {2}\, t}{2}\right )+5 \sqrt {2}\, \sin \left (\frac {\sqrt {2}\, t}{2}\right )\right ) \]

Problem 6676

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

Problem 6677

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=t +1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = -\frac {7 \,{\mathrm e}^{t} \cos \left (2 t \right )}{25}+\frac {51 \,{\mathrm e}^{t} \sin \left (2 t \right )}{25}+\frac {t}{5}+\frac {7}{25} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {7 \,{\mathrm e}^{t} \cos \left (2 t \right )}{25}+\frac {51 \,{\mathrm e}^{t} \sin \left (2 t \right )}{25}+\frac {t}{5}+\frac {7}{25} \]

Problem 6678

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

Problem 6679

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+20 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (\pi \right ) = 0] \end {align*}
program solution	\[ y = 0 \] Verified OK.
Maple solution	\[ y \left (t \right ) = 0 \]

Problem 6680

ODE	\[ \boxed {y^{\prime }+y=\left \{\begin {array}{cc} 0 & 0\le t <1 \\ 5 & 1\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ 5-5 \,{\mathrm e}^{-t +1} & 1\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left \{\begin {array}{cc} 0 & t <1 \\ 5-5 \,{\mathrm e}^{1-t} & 1\le t \end {array}\right . \]

Problem 6681

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

Problem 6682

ODE	\[ \boxed {y^{\prime }+y=\left \{\begin {array}{cc} t & 0\le t <1 \\ 0 & 1\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} -1+{\mathrm e}^{-t}+t & t <1 \\ 1+{\mathrm e}^{-1} & t =1 \\ -1+{\mathrm e}^{-t}+{\mathrm e}^{-t +1}+2 \,{\mathrm e}^{\frac {1}{2}-\frac {t}{2}} \sinh \left (-\frac {1}{2}+\frac {t}{2}\right ) & 1
Maple solution	\[ y \left (t \right ) = \left \{\begin {array}{cc} -1+{\mathrm e}^{-t}+t & t <1 \\ 1+{\mathrm e}^{-1} & t =1 \\ {\mathrm e}^{-t} & 1

Problem 6683

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

Problem 6684

ODE	\[ \boxed {y^{\prime \prime }+4 y=\sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \cos \left (2 t \right )+\left (\left \{\begin {array}{cc} 0 & t <2 \pi \\ \frac {\sin \left (t \right )}{3}-\frac {\sin \left (2 t \right )}{6} & 2 \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (\cos \left (t \right )-1\right ) \sin \left (t \right ) \operatorname {Heaviside}\left (t -2 \pi \right )}{3}+2 \cos \left (t \right )^{2}-1 \]

Problem 6685

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=\operatorname {Heaviside}\left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} -{\mathrm e}^{2 t}+{\mathrm e}^{3 t} & t <1 \\ -{\mathrm e}^{2}+{\mathrm e}^{3}+\frac {1}{6} & t =1 \\ -{\mathrm e}^{2 t}+{\mathrm e}^{3 t}+\frac {{\mathrm e}^{3 t -3}}{3}-\frac {{\mathrm e}^{2 t -2}}{2}+\frac {1}{6} & 1
Maple solution	\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -1\right )}{6}-{\mathrm e}^{2 t}+{\mathrm e}^{3 t}-\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{2 t -2}}{2}+\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{3 t -3}}{3} \]

Problem 6686

ODE	\[ \boxed {y^{\prime \prime }+y=\left \{\begin {array}{cc} 0 & 0\le t <\pi \\ 1 & \pi \le t <2 \pi \\ 0 & 2 \pi \le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \sin \left (t \right )+\left (\left \{\begin {array}{cc} 0 & t <\pi \\ 1+\cos \left (t \right ) & t <2 \pi \\ 2 \cos \left (t \right ) & 2 \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (t \right )+\left (\left \{\begin {array}{cc} 0 & t <\pi \\ \cos \left (t \right )+1 & t <2 \pi \\ 2 \cos \left (t \right ) & 2 \pi \le t \end {array}\right .\right ) \]

Problem 6687

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+3 y=1-\operatorname {Heaviside}\left (-2+t \right )-\operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -6\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left (\left \{\begin {array}{cc} {\mathrm e}^{-3 t}-3 \,{\mathrm e}^{-t}+2 & t <2 \\ {\mathrm e}^{-6}-3 \,{\mathrm e}^{-2}+4 & t =2 \\ {\mathrm e}^{-3 t}-3 \,{\mathrm e}^{-t}-{\mathrm e}^{6-3 t}+3 \,{\mathrm e}^{2-t} & t <4 \\ {\mathrm e}^{-3 t}-3 \,{\mathrm e}^{-t}-{\mathrm e}^{6-3 t}+3 \,{\mathrm e}^{2-t}-2-{\mathrm e}^{-3 t +12}+3 \,{\mathrm e}^{-t +4} & t <6 \\ {\mathrm e}^{-3 t}-3 \,{\mathrm e}^{-t}-{\mathrm e}^{6-3 t}+3 \,{\mathrm e}^{2-t}+{\mathrm e}^{-3 t +18}-3 \,{\mathrm e}^{-t +6}-{\mathrm e}^{-3 t +12}+3 \,{\mathrm e}^{-t +4} & 6\le t \end {array}\right .\right )}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {1}{3}-\frac {\operatorname {Heaviside}\left (t -4\right )}{3}+\frac {{\mathrm e}^{-3 t}}{6}-\frac {{\mathrm e}^{-t}}{2}-\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-t +6}}{2}+\frac {\operatorname {Heaviside}\left (t -6\right ) {\mathrm e}^{-3 t +18}}{6}+\frac {\operatorname {Heaviside}\left (t -6\right )}{3}+\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-t +4}}{2}-\frac {\operatorname {Heaviside}\left (t -4\right ) {\mathrm e}^{-3 t +12}}{6}+\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-t +2}}{2}-\frac {\operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-3 t +6}}{6}-\frac {\operatorname {Heaviside}\left (t -2\right )}{3} \]

Problem 6688

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

Problem 6689

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

Problem 6690

ODE	\[ \boxed {y^{\prime \prime }+9 y=\cos \left (3 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = 2 \cos \left (3 t \right )+\frac {\sin \left (3 t \right ) \left (10+t \right )}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 \cos \left (3 t \right )+\frac {\sin \left (3 t \right ) \left (10+t \right )}{6} \]

Problem 6691

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

Problem 6692

ODE	\[ \boxed {y^{\prime \prime }+16 y=\left \{\begin {array}{cc} \cos \left (4 t \right ) & 0\le t <\pi \\ 0 & \pi \le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {\sin \left (4 t \right ) \left (2+\left (\left \{\begin {array}{cc} t & t <\pi \\ \pi & \pi \le t \end {array}\right .\right )\right )}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\sin \left (4 t \right ) \left (2+\left (\left \{\begin {array}{cc} t & t <\pi \\ \pi & \pi \le t \end {array}\right .\right )\right )}{8} \]

Problem 6693

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

Problem 6694

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

Problem 6695

ODE	\[ \boxed {2 y^{\prime \prime }+y^{\prime } t -2 y=10} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (t \right ) = \frac {5 t^{2}}{2} \]

Problem 6696

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

Problem 6697

ODE	\[ \boxed {y^{\prime }-3 y=\delta \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <2 \\ {\mathrm e}^{-6+3 t} & 2\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{3 t -6} \]

Problem 6698

ODE	\[ \boxed {y^{\prime }+y=\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 2 \,{\mathrm e}^{-t} & t <1 \\ 2 \,{\mathrm e}^{-t}+{\mathrm e}^{-t +1} & 1\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t}+2 \,{\mathrm e}^{-t} \]

Problem 6699

ODE	\[ \boxed {y^{\prime \prime }+y=\delta \left (t -2 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \sin \left (t \right ) \left (\left \{\begin {array}{cc} 1 & t <2 \pi \\ 2 & 2 \pi \le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (t \right ) \left (\operatorname {Heaviside}\left (t -2 \pi \right )+1\right ) \]

Problem 6700

ODE	\[ \boxed {y^{\prime \prime }+16 y=\delta \left (t -2 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <2 \pi \\ \frac {\sin \left (4 t \right )}{4} & 2 \pi \le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -2 \pi \right ) \sin \left (4 t \right )}{4} \]