2.17.71 Problems 7001 to 7100

Problem 7001

ODE	\[ \boxed {x \left (x -2\right )^{2} y^{\prime \prime }-2 \left (x -2\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = c_{1} \left (x -2\right ) \left (1+O\left (\left (x -2\right )^{8}\right )\right )+c_{2} \left (\left (x -2\right ) \left (1+O\left (\left (x -2\right )^{8}\right )\right ) \ln \left (x -2\right )+\left (x -2\right ) \left (-\frac {x}{2}+1+\frac {\left (x -2\right )^{2}}{8}-\frac {\left (x -2\right )^{3}}{24}+\frac {\left (x -2\right )^{4}}{64}-\frac {\left (x -2\right )^{5}}{160}+\frac {\left (x -2\right )^{6}}{384}-\frac {\left (x -2\right )^{7}}{896}+O\left (\left (x -2\right )^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (\ln \left (-2+x \right ) c_{2} +c_{1} \right ) \left (1+\operatorname {O}\left (\left (-2+x \right )^{8}\right )\right )+\left (-\frac {1}{2} \left (-2+x \right )+\frac {1}{8} \left (-2+x \right )^{2}-\frac {1}{24} \left (-2+x \right )^{3}+\frac {1}{64} \left (-2+x \right )^{4}-\frac {1}{160} \left (-2+x \right )^{5}+\frac {1}{384} \left (-2+x \right )^{6}-\frac {1}{896} \left (-2+x \right )^{7}+\operatorname {O}\left (\left (-2+x \right )^{8}\right )\right ) c_{2} \right ) \left (-2+x \right ) \]

Problem 7002

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }-\left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {9 x^{2}}{40}+\frac {103 x^{3}}{1680}+\frac {187 x^{4}}{13440}+\frac {247 x^{5}}{98560}+\frac {17861 x^{6}}{46126080}+\frac {23767 x^{7}}{461260800}+O\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x +\frac {9}{40} x^{2}+\frac {103}{1680} x^{3}+\frac {187}{13440} x^{4}+\frac {247}{98560} x^{5}+\frac {17861}{46126080} x^{6}+\frac {23767}{461260800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 7003

ODE	\[ \boxed {x y^{\prime \prime }-\left (x +2\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right )+c_{2} \left (\frac {x^{3} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{2}+1-\frac {x}{2}+\frac {x^{2}}{2}-\frac {x^{4}}{2}-\frac {3 x^{5}}{8}-\frac {11 x^{6}}{72}-\frac {25 x^{7}}{576}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (6 x^{3}+6 x^{4}+3 x^{5}+x^{6}+\frac {1}{4} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (12-6 x +6 x^{2}+11 x^{3}+5 x^{4}+x^{5}-\frac {1}{16} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 7004

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	\[ y = c_{1} x^{3} \left (1+\frac {5 x}{4}+\frac {3 x^{2}}{4}+\frac {7 x^{3}}{24}+\frac {x^{4}}{12}+\frac {3 x^{5}}{160}+\frac {x^{6}}{288}+\frac {11 x^{7}}{20160}+O\left (x^{8}\right )\right )+c_{2} \left (2 x^{3} \left (1+\frac {5 x}{4}+\frac {3 x^{2}}{4}+\frac {7 x^{3}}{24}+\frac {x^{4}}{12}+\frac {3 x^{5}}{160}+\frac {x^{6}}{288}+\frac {11 x^{7}}{20160}+O\left (x^{8}\right )\right ) \ln \left (x \right )+1-x +\frac {3 x^{2}}{2}-\frac {21 x^{4}}{8}-\frac {19 x^{5}}{8}-\frac {163 x^{6}}{144}-\frac {53 x^{7}}{144}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+\frac {5}{4} x +\frac {3}{4} x^{2}+\frac {7}{24} x^{3}+\frac {1}{12} x^{4}+\frac {3}{160} x^{5}+\frac {1}{288} x^{6}+\frac {11}{20160} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (24 x^{3}+30 x^{4}+18 x^{5}+7 x^{6}+2 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (12-12 x +18 x^{2}+26 x^{3}+x^{4}-9 x^{5}-6 x^{6}-\frac {9}{4} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

Problem 7005

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x^{2}-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-x +\frac {3 x^{2}}{5}-\frac {4 x^{3}}{15}+\frac {2 x^{4}}{21}-\frac {x^{5}}{35}+\frac {x^{6}}{135}-\frac {8 x^{7}}{4725}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-x +\frac {2 x^{3}}{3}-\frac {2 x^{4}}{3}+\frac {2 x^{5}}{5}-\frac {8 x^{6}}{45}+\frac {4 x^{7}}{63}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1-x +\frac {3}{5} x^{2}-\frac {4}{15} x^{3}+\frac {2}{21} x^{4}-\frac {1}{35} x^{5}+\frac {1}{135} x^{6}-\frac {8}{4725} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-12 x +8 x^{3}-8 x^{4}+\frac {24}{5} x^{5}-\frac {32}{15} x^{6}+\frac {16}{21} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 7006

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-x \left (2 x +7\right ) y^{\prime }+2 \left (x +5\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {5}{2}} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right )+c_{2} x^{2} \left (1+2 x +\frac {4 x^{2}}{3}+\frac {8 x^{3}}{15}+\frac {16 x^{4}}{105}+\frac {32 x^{5}}{945}+\frac {64 x^{6}}{10395}+\frac {128 x^{7}}{135135}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+2 x +\frac {4}{3} x^{2}+\frac {8}{15} x^{3}+\frac {16}{105} x^{4}+\frac {32}{945} x^{5}+\frac {64}{10395} x^{6}+\frac {128}{135135} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\frac {5}{2}} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 7007

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

Problem 7008

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }-18 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (135 x^{8}+70 x^{6}+30 x^{4}+9 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {14}{3} x^{3}+\frac {63}{5} x^{5}+\frac {132}{5} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (70 x^{6}+30 x^{4}+9 x^{2}+1\right ) c_{1} +\left (x +\frac {14}{3} x^{3}+\frac {63}{5} x^{5}+\frac {132}{5} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (70 x^{6}+30 x^{4}+9 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {14}{3} x^{3}+\frac {63}{5} x^{5}+\frac {132}{5} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 7009

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (1+2 x \right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {2 x}{3}+O\left (x^{8}\right )\right )+c_{2} \left (1+3 x +\frac {x^{2}}{2}-\frac {x^{3}}{30}+\frac {x^{4}}{280}-\frac {x^{5}}{2520}+\frac {x^{6}}{23760}-\frac {x^{7}}{240240}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {2}{3} x +\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (1+3 x +\frac {1}{2} x^{2}-\frac {1}{30} x^{3}+\frac {1}{280} x^{4}-\frac {1}{2520} x^{5}+\frac {1}{23760} x^{6}-\frac {1}{240240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 7010

ODE	\[ \boxed {y^{\prime \prime }+2 x y^{\prime }-8 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+4 x^{2}+\frac {4}{3} x^{4}\right ) y \left (0\right )+\left (x +x^{3}+\frac {1}{10} x^{5}-\frac {1}{210} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (1+4 x^{2}+\frac {4}{3} x^{4}\right ) c_{1} +\left (x +x^{3}+\frac {1}{10} x^{5}-\frac {1}{210} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\frac {4}{3} x^{4}+4 x^{2}+1\right ) y \left (0\right )+\left (x +x^{3}+\frac {1}{10} x^{5}-\frac {1}{210} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 7011

ODE	\[ \boxed {x \left (-x^{2}+1\right ) y^{\prime \prime }-\left (x^{2}+7\right ) y^{\prime }+4 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{8} \left (1+3 x^{2}+6 x^{4}+10 x^{6}+15 x^{8}+O\left (x^{9}\right )\right )+c_{2} \left (1+\frac {x^{2}}{3}+\frac {x^{8}}{3}+O\left (x^{9}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{8} \left (1+3 x^{2}+6 x^{4}+10 x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-203212800-67737600 x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 7012

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+\left (1+4 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {2 x}{3}+O\left (x^{8}\right )\right )+c_{2} \sqrt {x}\, \left (1-3 x +\frac {x^{2}}{2}+\frac {x^{3}}{30}+\frac {x^{4}}{280}+\frac {x^{5}}{2520}+\frac {x^{6}}{23760}+\frac {x^{7}}{240240}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-3 x +\frac {1}{2} x^{2}+\frac {1}{30} x^{3}+\frac {1}{280} x^{4}+\frac {1}{2520} x^{5}+\frac {1}{23760} x^{6}+\frac {1}{240240} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {2}{3} x +\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 7013

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-2 x \left (x +2\right ) y^{\prime }+\left (x +3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1+\frac {x}{4}+\frac {x^{2}}{24}+\frac {x^{3}}{192}+\frac {x^{4}}{1920}+\frac {x^{5}}{23040}+\frac {x^{6}}{322560}+\frac {x^{7}}{5160960}+O\left (x^{8}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x}{2}+\frac {x^{2}}{8}+\frac {x^{3}}{48}+\frac {x^{4}}{384}+\frac {x^{5}}{3840}+\frac {x^{6}}{46080}+\frac {x^{7}}{645120}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (x \left (1+\frac {1}{4} x +\frac {1}{24} x^{2}+\frac {1}{192} x^{3}+\frac {1}{1920} x^{4}+\frac {1}{23040} x^{5}+\frac {1}{322560} x^{6}+\frac {1}{5160960} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{1} +\left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\frac {1}{46080} x^{6}+\frac {1}{645120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

Problem 7014

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (x^{2}+1\right ) y^{\prime }+\left (-x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{6}}{48}+O\left (x^{8}\right )\right )+c_{2} \left (x \left (1+\frac {x^{2}}{2}+\frac {x^{4}}{8}+\frac {x^{6}}{48}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x \left (-\frac {x^{2}}{4}-\frac {3 x^{4}}{32}-\frac {11 x^{6}}{576}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{2} x^{2}+\frac {1}{8} x^{4}+\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}-\frac {3}{32} x^{4}-\frac {11}{576} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

Problem 7015

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 7016

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x^{2}-3\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{8}-\frac {x^{6}}{48}+O\left (x^{8}\right )\right )+c_{2} \left (x^{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{8}-\frac {x^{6}}{48}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x^{2} \left (\frac {x^{2}}{4}-\frac {3 x^{4}}{32}+\frac {11 x^{6}}{576}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{32} x^{4}+\frac {11}{576} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

Problem 7017

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-y^{\prime } x^{2}+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1+\frac {x}{8}+\frac {3 x^{2}}{256}+\frac {5 x^{3}}{6144}+\frac {35 x^{4}}{786432}+\frac {21 x^{5}}{10485760}+\frac {77 x^{6}}{1006632960}+\frac {143 x^{7}}{56371445760}+O\left (x^{8}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1+\frac {x}{8}+\frac {3 x^{2}}{256}+\frac {5 x^{3}}{6144}+\frac {35 x^{4}}{786432}+\frac {21 x^{5}}{10485760}+\frac {77 x^{6}}{1006632960}+\frac {143 x^{7}}{56371445760}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (-\frac {x^{2}}{256}-\frac {x^{3}}{2048}-\frac {19 x^{4}}{524288}-\frac {25 x^{5}}{12582912}-\frac {317 x^{6}}{3623878656}-\frac {469 x^{7}}{144955146240}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{8} x +\frac {3}{256} x^{2}+\frac {5}{6144} x^{3}+\frac {35}{786432} x^{4}+\frac {21}{10485760} x^{5}+\frac {77}{1006632960} x^{6}+\frac {143}{56371445760} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{256} x^{2}-\frac {1}{2048} x^{3}-\frac {19}{524288} x^{4}-\frac {25}{12582912} x^{5}-\frac {317}{3623878656} x^{6}-\frac {469}{144955146240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

Problem 7018

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (x^{2}+1\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {1}{15} x^{5}+\frac {1}{35} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \] Verified OK. \[ y = \left (x^{2}+1\right ) c_{1} +\left (x +\frac {1}{3} x^{3}-\frac {1}{15} x^{5}+\frac {1}{35} x^{7}\right ) c_{2} +O\left (x^{8}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{2}+1\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {1}{15} x^{5}+\frac {1}{35} x^{7}\right ) D\left (y \right )\left (0\right )+O\left (x^{8}\right ) \]

Problem 7019

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-x \left (1+2 x \right ) y^{\prime }+\left (1+3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x \left (1-\frac {x}{3}-\frac {x^{2}}{30}-\frac {x^{3}}{210}-\frac {x^{4}}{1512}-\frac {x^{5}}{11880}-\frac {x^{6}}{102960}-\frac {x^{7}}{982800}+O\left (x^{8}\right )\right )+c_{2} \sqrt {x}\, \left (1-2 x +O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-2 x +\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1-\frac {1}{3} x -\frac {1}{30} x^{2}-\frac {1}{210} x^{3}-\frac {1}{1512} x^{4}-\frac {1}{11880} x^{5}-\frac {1}{102960} x^{6}-\frac {1}{982800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 7020

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+3 y^{\prime } x^{2}+\left (1+3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {9 x}{8}+\frac {135 x^{2}}{256}-\frac {315 x^{3}}{2048}+\frac {8505 x^{4}}{262144}-\frac {56133 x^{5}}{10485760}+\frac {243243 x^{6}}{335544320}-\frac {312741 x^{7}}{3758096384}+O\left (x^{8}\right )\right )+c_{2} \left (\sqrt {x}\, \left (1-\frac {9 x}{8}+\frac {135 x^{2}}{256}-\frac {315 x^{3}}{2048}+\frac {8505 x^{4}}{262144}-\frac {56133 x^{5}}{10485760}+\frac {243243 x^{6}}{335544320}-\frac {312741 x^{7}}{3758096384}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\sqrt {x}\, \left (\frac {3 x}{2}-\frac {261 x^{2}}{256}+\frac {729 x^{3}}{2048}-\frac {44091 x^{4}}{524288}+\frac {63099 x^{5}}{4194304}-\frac {1454463 x^{6}}{671088640}+\frac {1403811 x^{7}}{5368709120}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {9}{8} x +\frac {135}{256} x^{2}-\frac {315}{2048} x^{3}+\frac {8505}{262144} x^{4}-\frac {56133}{10485760} x^{5}+\frac {243243}{335544320} x^{6}-\frac {312741}{3758096384} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{2} x -\frac {261}{256} x^{2}+\frac {729}{2048} x^{3}-\frac {44091}{524288} x^{4}+\frac {63099}{4194304} x^{5}-\frac {1454463}{671088640} x^{6}+\frac {1403811}{5368709120} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

Problem 7021

ODE	\[ \boxed {x y^{\prime \prime }+\left (-x^{2}+1\right ) y^{\prime }+2 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{2}+O\left (x^{8}\right )\right )+c_{2} \left (\left (1-\frac {x^{2}}{2}+O\left (x^{8}\right )\right ) \ln \left (x \right )+\frac {3 x^{2}}{4}-\frac {x^{4}}{32}-\frac {x^{6}}{576}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{2} x^{2}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{4} x^{2}-\frac {1}{32} x^{4}-\frac {1}{576} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

Problem 7022

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+2 y^{\prime } x^{2}-\left (x +3\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x}{6}+\frac {x^{2}}{48}-\frac {x^{3}}{480}+\frac {x^{4}}{5760}-\frac {x^{5}}{80640}+\frac {x^{6}}{1290240}-\frac {x^{7}}{23224320}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {x}{2}+\frac {x^{2}}{8}-\frac {x^{3}}{48}+\frac {x^{4}}{384}-\frac {x^{5}}{3840}+\frac {x^{6}}{46080}-\frac {x^{7}}{645120}+O\left (x^{8}\right )\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{2} \left (1-\frac {1}{6} x +\frac {1}{48} x^{2}-\frac {1}{480} x^{3}+\frac {1}{5760} x^{4}-\frac {1}{80640} x^{5}+\frac {1}{1290240} x^{6}-\frac {1}{23224320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (-2+x -\frac {1}{4} x^{2}+\frac {1}{24} x^{3}-\frac {1}{192} x^{4}+\frac {1}{1920} x^{5}-\frac {1}{23040} x^{6}+\frac {1}{322560} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]

Problem 7023

ODE	\[ \boxed {x \left (-x^{2}+1\right ) y^{\prime \prime }+5 \left (-x^{2}+1\right ) y^{\prime }-4 y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x^{2}}{3}+\frac {x^{4}}{6}+\frac {x^{6}}{10}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1-x^{2}+O\left (x^{8}\right )\right )}{x^{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {1}{3} x^{2}+\frac {1}{6} x^{4}+\frac {1}{10} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-144+144 x^{2}+\operatorname {O}\left (x^{8}\right )\right )}{x^{4}} \]

Problem 7024

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

Problem 7025

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }-\left (x^{2}+4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+\frac {x^{2}}{12}+\frac {x^{4}}{384}+\frac {x^{6}}{23040}+O\left (x^{8}\right )\right )+c_{2} \left (-\frac {x^{2} \left (1+\frac {x^{2}}{12}+\frac {x^{4}}{384}+\frac {x^{6}}{23040}+O\left (x^{8}\right )\right ) \ln \left (x \right )}{16}+\frac {1-\frac {x^{2}}{4}+\frac {x^{6}}{288}+O\left (x^{8}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{4} \left (1+\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\frac {1}{23040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\frac {3}{4} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (-144+36 x^{2}-\frac {1}{2} x^{6}+\operatorname {O}\left (x^{8}\right )\right )\right )}{x^{2}} \]

Problem 7026

ODE	\[ \boxed {x \left (1-2 x \right ) y^{\prime \prime }-2 \left (x +2\right ) y^{\prime }+18 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{5} \left (1+\frac {16 x}{3}+\frac {144 x^{2}}{7}+\frac {480 x^{3}}{7}+\frac {4400 x^{4}}{21}+\frac {4224 x^{5}}{7}+1664 x^{6}+\frac {13312 x^{7}}{3}+O\left (x^{8}\right )\right )+c_{2} \left (1+\frac {9 x}{2}+12 x^{2}+20 x^{3}-168 x^{5}-896 x^{6}-3456 x^{7}+O\left (x^{8}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{5} \left (1+\frac {16}{3} x +\frac {144}{7} x^{2}+\frac {480}{7} x^{3}+\frac {4400}{21} x^{4}+\frac {4224}{7} x^{5}+1664 x^{6}+\frac {13312}{3} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (2880+12960 x +34560 x^{2}+57600 x^{3}-483840 x^{5}-2580480 x^{6}-9953280 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

Problem 7027

ODE	\[ \boxed {x y^{\prime \prime }+\left (-x +2\right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \left (1+\frac {x}{2}+\frac {x^{2}}{6}+\frac {x^{3}}{24}+\frac {x^{4}}{120}+\frac {x^{5}}{720}+\frac {x^{6}}{5040}+\frac {x^{7}}{40320}+O\left (x^{8}\right )\right )+\frac {c_{2} \left (1+x +\frac {x^{2}}{2}+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {x^{5}}{120}+\frac {x^{6}}{720}+\frac {x^{7}}{5040}+O\left (x^{8}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1+\frac {1}{2} x +\frac {1}{6} x^{2}+\frac {1}{24} x^{3}+\frac {1}{120} x^{4}+\frac {1}{720} x^{5}+\frac {1}{5040} x^{6}+\frac {1}{40320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

Problem 7028

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 x y^{\prime }+4 \left (1+x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (4 x^{2}-4 x +1-\frac {16 x^{3}}{9}+\frac {4 x^{4}}{9}-\frac {16 x^{5}}{225}+\frac {16 x^{6}}{2025}-\frac {64 x^{7}}{99225}+O\left (x^{8}\right )\right )+c_{2} \left (x^{2} \left (4 x^{2}-4 x +1-\frac {16 x^{3}}{9}+\frac {4 x^{4}}{9}-\frac {16 x^{5}}{225}+\frac {16 x^{6}}{2025}-\frac {64 x^{7}}{99225}+O\left (x^{8}\right )\right ) \ln \left (x \right )+x^{2} \left (-12 x^{2}+8 x +\frac {176 x^{3}}{27}-\frac {50 x^{4}}{27}+\frac {1096 x^{5}}{3375}-\frac {392 x^{6}}{10125}+\frac {3872 x^{7}}{1157625}+O\left (x^{8}\right )\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-4 x +4 x^{2}-\frac {16}{9} x^{3}+\frac {4}{9} x^{4}-\frac {16}{225} x^{5}+\frac {16}{2025} x^{6}-\frac {64}{99225} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (8 x -12 x^{2}+\frac {176}{27} x^{3}-\frac {50}{27} x^{4}+\frac {1096}{3375} x^{5}-\frac {392}{10125} x^{6}+\frac {3872}{1157625} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) x^{2} \]

Problem 7029

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

Problem 7030

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

Problem 7031

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

Problem 7032

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

Problem 7033

ODE	\[ \boxed {y^{\prime }-\frac {2 x -y}{x +4 y}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ -\left (x +y\right ) \left (x -2 y\right ) = 2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{4}+\frac {\sqrt {9 x^{2}+16}}{4} \]

Problem 7034

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

Problem 7035

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

Problem 7036

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

Problem 7037

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

Problem 7038

ODE	\[ \boxed {y^{\prime \prime }+16 y=4 \cos \left (x \right )} \]
program solution	\[ y = c_{1} \cos \left (4 x \right )+\frac {c_{2} \sin \left (4 x \right )}{4}+\frac {4 \cos \left (x \right )}{15} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (4 x \right ) c_{2} +\cos \left (4 x \right ) c_{1} +\frac {4 \cos \left (x \right )}{15} \]

Problem 7039

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=9 x^{2}+4} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = 8] \end {align*}
program solution	\[ y = 10+3 x^{2}+2 \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{x}+8 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{3 x}-6 \,{\mathrm e}^{x}+3 x^{2}+8 x +10 \]

Problem 7040

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

Problem 7041

ODE	\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )+5 y \left (t \right ) \end {align*} With initial conditions \[ [x \left (0\right ) = -2, y \left (0\right ) = 1] \]
program solution
Maple solution	\begin{align*} x \left (t \right ) &= -3 \,{\mathrm e}^{-t}+{\mathrm e}^{4 t} \\ y \left (t \right ) &= -{\mathrm e}^{-t}+2 \,{\mathrm e}^{4 t} \\ \end{align*}

Problem 7042

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

Problem 7043

ODE	\begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right )+4 \,{\mathrm e}^{t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{1} {\mathrm e}^{3 t}+2 \,{\mathrm e}^{t} t \\ y \left (t \right ) &= c_{2} {\mathrm e}^{t}-c_{1} {\mathrm e}^{3 t}+2 \,{\mathrm e}^{t} t -2 \,{\mathrm e}^{t} \\ \end{align*}

Problem 7044

ODE	\begin {align*} x^{\prime }\left (t \right )&=6 x \left (t \right )-7 y \left (t \right )+10\\ y^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right )-2 \,{\mathrm e}^{t} \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{5 t}+{\mathrm e}^{-t} c_{1} -\frac {7 \,{\mathrm e}^{t}}{4}-4 \\ y \left (t \right ) &= \frac {c_{2} {\mathrm e}^{5 t}}{7}+{\mathrm e}^{-t} c_{1} -\frac {5 \,{\mathrm e}^{t}}{4}-2 \\ \end{align*}

Problem 7045

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

Problem 7046

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

Problem 7047

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

Problem 7048

ODE	\[ \boxed {y^{\prime }-\left (5+\frac {\sec \left (x \right )}{x}\right ) \left (\sin \left (y\right )+y\right )=0} \]
program solution	\[ \int _{}^{x}-\frac {\sec \left (\textit {\_a} \right )+5 \textit {\_a}}{\textit {\_a}}d \textit {\_a} +\int _{0}^{y}\frac {1}{\sin \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} = c_{1} \] Verified OK.
Maple solution	\[ \int \frac {5 x +\sec \left (x \right )}{x}d x -\left (\int _{}^{y \left (x \right )}\frac {1}{\sin \left (\textit {\_a} \right )+\textit {\_a}}d \textit {\_a} \right )+c_{1} = 0 \]

Problem 7049

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

Problem 7050

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

Problem 7051

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

Problem 7052

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

Problem 7053

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

Problem 7054

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

Problem 7055

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

Problem 7056

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

Problem 7057

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

Problem 7058

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

Problem 7059

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

Problem 7060

ODE	\[ \boxed {y^{\prime }-\frac {-y x -1}{4 x^{3} y-2 x^{2}}=0} \]
program solution	\[ \frac {\left (4 y x +1\right )^{\frac {3}{5}} \left (y x -1\right )^{\frac {2}{5}} 2^{\frac {4}{5}}}{4 x} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\textit {\_Z}^{25} c_{1} -10 \textit {\_Z}^{20} c_{1} +25 \textit {\_Z}^{15} c_{1} -16 x^{5}\right )^{5}-1}{4 x} \]

Problem 7061

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

Problem 7062

ODE	\[ \boxed {y^{\prime }-\sqrt {\frac {y+1}{y^{2}}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ \frac {2 y \left (-2+y\right ) \sqrt {\frac {y+1}{y^{2}}}}{3} = x -\frac {2 \sqrt {2}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (1+i \sqrt {3}\right ) \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{\frac {1}{3}}+4}{4 \left (-12 \sqrt {2}\, x +9 x^{2}+\sqrt {\left (-12 \sqrt {2}\, x +9 x^{2}-8\right ) \left (3 x -2 \sqrt {2}\right )^{2}}\right )^{\frac {1}{3}}} \]

Problem 7063

ODE	\[ \boxed {y^{\prime }-\sqrt {1-x^{2}-y^{2}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 7064

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

Problem 7065

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

Problem 7066

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

Problem 7067

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

Problem 7068

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

Problem 7069

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

Problem 7070

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

Problem 7071

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

Problem 7072

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

Problem 7073

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

Problem 7074

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

Problem 7075

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

Problem 7076

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

Problem 7077

ODE	\[ \boxed {p^{\prime }-a p+b p^{2}=0} \] With initial conditions \begin {align*} [p \left (\operatorname {t0} \right ) = \operatorname {p0}] \end {align*}
program solution	\[ \frac {\ln \left (p\right )-\ln \left (b p-a \right )}{a} = \frac {-\ln \left (b \operatorname {p0} -a \right )+\ln \left (\operatorname {p0} \right )+\left (t -\operatorname {t0} \right ) a}{a} \] Verified OK.
Maple solution	\[ p \left (t \right ) = \frac {a \operatorname {p0}}{\left (-\operatorname {p0} b +a \right ) {\mathrm e}^{-a \left (t -\operatorname {t0} \right )}+\operatorname {p0} b} \]

Problem 7078

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

Problem 7079

ODE	\[ \boxed {x f^{\prime }-f-\frac {{f^{\prime }}^{2} \left (1-{f^{\prime }}^{\lambda }\right )^{2}}{\lambda ^{2}}=0} \]
program solution	\[ f = c_{1} x -\frac {c_{1}^{2} \left (c_{1}^{2 \lambda }-2 c_{1}^{\lambda }+1\right )}{\lambda ^{2}} \] Verified OK. \[ f = \frac {\operatorname {RootOf}\left (2 \textit {\_Z}^{1+2 \lambda } \lambda +2 \textit {\_Z}^{1+2 \lambda }-2 \textit {\_Z}^{\lambda +1} \lambda -x \,\lambda ^{2}-4 \textit {\_Z}^{\lambda +1}+2 \textit {\_Z} \right ) x \,\lambda ^{2}+2 \operatorname {RootOf}\left (2 \textit {\_Z}^{1+2 \lambda } \lambda +2 \textit {\_Z}^{1+2 \lambda }-2 \textit {\_Z}^{\lambda +1} \lambda -x \,\lambda ^{2}-4 \textit {\_Z}^{\lambda +1}+2 \textit {\_Z} \right )^{\lambda +2}-\operatorname {RootOf}\left (2 \textit {\_Z}^{1+2 \lambda } \lambda +2 \textit {\_Z}^{1+2 \lambda }-2 \textit {\_Z}^{\lambda +1} \lambda -x \,\lambda ^{2}-4 \textit {\_Z}^{\lambda +1}+2 \textit {\_Z} \right )^{2}-\operatorname {RootOf}\left (2 \textit {\_Z}^{1+2 \lambda } \lambda +2 \textit {\_Z}^{1+2 \lambda }-2 \textit {\_Z}^{\lambda +1} \lambda -x \,\lambda ^{2}-4 \textit {\_Z}^{\lambda +1}+2 \textit {\_Z} \right )^{2 \lambda +2}}{\lambda ^{2}} \] Warning, solution could not be verified
Maple solution	\begin{align*} f \left (x \right ) &= 0 \\ f \left (x \right ) &= \frac {\lambda ^{2} x^{2} \left (2 \lambda \,{\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }+{\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }-1\right )}{4 \left (\lambda \,{\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }+{\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }-1\right )^{2} \left ({\mathrm e}^{\operatorname {RootOf}\left (2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z} \left (2 \lambda +1\right )}-2 \lambda \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}-x \,\lambda ^{2}-4 \,{\mathrm e}^{\textit {\_Z} \left (\lambda +1\right )}+2 \,{\mathrm e}^{\textit {\_Z}}\right ) \lambda }-1\right )} \\ f \left (x \right ) &= c_{1} x -\frac {c_{1}^{2} \left (-1+c_{1}^{\lambda }\right )^{2}}{\lambda ^{2}} \\ \end{align*}

Problem 7080

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

Problem 7081

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

Problem 7082

ODE	\[ \boxed {u^{\prime }+u^{2}=\frac {1}{x^{\frac {4}{5}}}} \]
program solution	\[ u = \frac {-\operatorname {BesselK}\left (\frac {1}{6}, \frac {5 x^{\frac {3}{5}}}{3}\right )+\operatorname {BesselI}\left (-\frac {1}{6}, \frac {5 x^{\frac {3}{5}}}{3}\right ) c_{3}}{x^{\frac {2}{5}} \left (\operatorname {BesselK}\left (\frac {5}{6}, \frac {5 x^{\frac {3}{5}}}{3}\right )+\operatorname {BesselI}\left (\frac {5}{6}, \frac {5 x^{\frac {3}{5}}}{3}\right ) c_{3} \right )} \] Verified OK.
Maple solution	\[ u \left (x \right ) = \frac {\operatorname {BesselI}\left (-\frac {1}{6}, \frac {5 x^{\frac {3}{5}}}{3}\right ) c_{1} -\operatorname {BesselK}\left (\frac {1}{6}, \frac {5 x^{\frac {3}{5}}}{3}\right )}{x^{\frac {2}{5}} \left (c_{1} \operatorname {BesselI}\left (\frac {5}{6}, \frac {5 x^{\frac {3}{5}}}{3}\right )+\operatorname {BesselK}\left (\frac {5}{6}, \frac {5 x^{\frac {3}{5}}}{3}\right )\right )} \]

Problem 7083

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

Problem 7084

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

Problem 7085

ODE	\[ \boxed {5 y^{\prime \prime }+2 y^{\prime }+4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = \frac {25 \sqrt {19}\, {\mathrm e}^{-\frac {x}{5}} \sin \left (\frac {\sqrt {19}\, x}{5}\right )}{19} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {25 \sqrt {19}\, {\mathrm e}^{-\frac {x}{5}} \sin \left (\frac {\sqrt {19}\, x}{5}\right )}{19} \]

Problem 7086

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+4 y=1} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {15}\, x}{2}\right )+\frac {2 c_{2} {\mathrm e}^{-\frac {x}{2}} \sqrt {15}\, \sin \left (\frac {\sqrt {15}\, x}{2}\right )}{15}+\frac {1}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {15}\, x}{2}\right ) c_{2} +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {15}\, x}{2}\right ) c_{1} +\frac {1}{4} \]

Problem 7087

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+4 y=\sin \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {15}\, x}{2}\right )+\frac {2 c_{2} {\mathrm e}^{-\frac {x}{2}} \sqrt {15}\, \sin \left (\frac {\sqrt {15}\, x}{2}\right )}{15}-\frac {\cos \left (x \right )}{10}+\frac {3 \sin \left (x \right )}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {15}\, x}{2}\right ) c_{2} +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {15}\, x}{2}\right ) c_{1} +\frac {3 \sin \left (x \right )}{10}-\frac {\cos \left (x \right )}{10} \]

Problem 7088

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

Problem 7089

ODE	\[ \boxed {y^{\prime } y+x {y^{\prime }}^{3}=1} \]
program solution	\[ y = -x -1 \] Verified OK. \[ x = \frac {54 x^{3} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\frac {x \left (\sqrt {\frac {4 y^{3}+27 x}{x}}\, c_{1} 3^{\frac {1}{6}}-2 \,3^{\frac {2}{3}} \left (y-\frac {3 c_{1}}{2}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}+\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+3 x 3^{\frac {1}{3}} \left (\frac {2 y^{2} c_{1}}{9}+x \right ) 2^{\frac {2}{3}}+{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (-\frac {4 y c_{1}}{3}+x \right )\right ) 2^{\frac {2}{3}}}{\left (-2^{\frac {2}{3}} 3^{\frac {1}{3}} x y+{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (-2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} x y+2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2}} \] Warning, solution could not be verified \[ x = -\frac {36 x^{3} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (-\frac {8 y c_{1}}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {\left (\left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) c_{1} \sqrt {\frac {4 y^{3}+27 x}{x}}-6 \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) \left (y-\frac {3 c_{1}}{2}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) 2^{\frac {2}{3}}}{{\left (\left (-i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) y 2^{\frac {2}{3}}\right )}^{2} {\left (\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2}} \] Warning, solution could not be verified \[ x = \frac {36 x^{3} \left (\left (\frac {8 y c_{1}}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {\left (\left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) c_{1} \sqrt {\frac {4 y^{3}+27 x}{x}}-6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) \left (y-\frac {3 c_{1}}{2}\right )\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {4 y^{3}+27 x}{x}}}{3}+\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) \left (\frac {2 y^{2} c_{1}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 2^{\frac {2}{3}}}{{\left (\frac {\left (3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\left (\left (-i+\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) y 2^{\frac {2}{3}}\right )}^{2}} \] Warning, solution could not be verified
Maple solution	\begin{align*} \frac {12 \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )+\left (\frac {3^{\frac {2}{3}} 2^{\frac {1}{3}} \left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{6}+2^{\frac {2}{3}} 3^{\frac {1}{3}} y \left (x \right )^{2}\right ) x \right ) c_{1} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} x^{3}}{\left (2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}-2 x \left (y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}}-3 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )\right )^{2} \left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x -{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}}+x -\frac {18 x^{4} \left (\sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}\, 2^{\frac {2}{3}} 3^{\frac {5}{6}} x -2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}} 3^{\frac {2}{3}} 2^{\frac {1}{3}} y \left (x \right )+9 \,3^{\frac {1}{3}} 2^{\frac {2}{3}} x +3 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{\left (-2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}+6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2} \left (-y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} &= 0 \\ -\frac {{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (-8 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )+x \left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}-2 y \left (x \right )^{2} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}\right )\right ) c_{1} x^{3}}{6 {\left (\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+\left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2} {\left (\left (-i+\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right ) x 2^{\frac {2}{3}}\right )}^{2}}+x +\frac {24 \,2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} x^{4} 3^{\frac {1}{3}} \left (-{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\frac {x \left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) 2^{\frac {2}{3}} \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}}{2}+\frac {3 x \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}\right )}{{\left (\left (-i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (-i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right ) x \right )}^{2} {\left (\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+\left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}} &= 0 \\ \frac {{\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} c_{1} x^{3} \left (8 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )+\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}-2 y \left (x \right )^{2} \left (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}}\right ) x \right )}{6 {\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) x 2^{\frac {2}{3}}\right )}^{2} {\left (\frac {\left (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+\left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}}+x +\frac {24 \,2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} x^{4} 3^{\frac {1}{3}} \left ({\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) 2^{\frac {2}{3}} \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}}{2}+\frac {3 x \left (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}}}{2}\right )}{{\left (\left (\sqrt {3}+i\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (3^{\frac {5}{6}}-i 3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} y \left (x \right ) x \right )}^{2} {\left (\frac {\left (-3^{\frac {1}{3}}+i 3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+\left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+y \left (x \right ) 2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x \right )}^{2}} &= 0 \\ \end{align*}

Problem 7090

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

Problem 7091

ODE	\[ \boxed {t y^{\prime \prime }+4 y^{\prime }=t^{2}} \]
program solution	\[ y = \frac {\left (t^{3}+3 c_{1} \right )^{2}}{18 t^{3}}+\frac {c_{2}}{t^{3}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{3}}{18}-\frac {c_{1}}{3 t^{3}}+c_{2} \]

Problem 7092

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

Problem 7093

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

Problem 7094

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

Problem 7095

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

Problem 7096

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

Problem 7097

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

Problem 7098

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

Problem 7099

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

Problem 7100

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