2.17.12 Problems 1101 to 1200

Problem 1101

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

Problem 1102

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

Problem 1103

ODE	\[ \boxed {\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }+\left (6 x -8\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = -\frac {6615 \,{\mathrm e}^{-\frac {x \left (i \sqrt {7}-1\right )}{2}} \left (x -\frac {1}{3}\right )^{2} \left (\left (\left (-\frac {23 x}{35}+\frac {296}{735}\right ) \sqrt {7}+i x -\frac {248 i}{105}\right ) c_{1} \operatorname {KummerM}\left (\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )-\left (\left (-\frac {x}{5}-\frac {8}{105}\right ) \sqrt {7}+i x -\frac {152 i}{105}\right ) c_{2} \operatorname {KummerU}\left (\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )+2 \left (i+\frac {3 \sqrt {7}}{49}\right ) c_{1} \operatorname {KummerM}\left (-\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right )+\frac {\left (i+\frac {5 \sqrt {7}}{7}\right ) \operatorname {KummerU}\left (-\frac {1}{2}-\frac {5 i \sqrt {7}}{14}, 3, \frac {i \sqrt {7}\, \left (3 x -1\right )}{3}\right ) c_{2}}{5}\right )}{-225 \sqrt {7}+21 i} \]

Problem 1104

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

Problem 1105

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

Problem 1106

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

Problem 1107

ODE	\[ \boxed {\left (1+2 x \right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y=\left (1+2 x \right )^{2}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} x +1-2 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{2} +x \,{\mathrm e}^{x} c_{1} +1-2 x \]

Problem 1108

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -y=\frac {4}{x^{2}}} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x -\frac {c_{2}}{2 x}+\frac {4}{3 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} x +\frac {4}{3 x^{2}}+\frac {c_{1}}{x} \]

Problem 1109

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +y=x} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = x \left (c_{1} +c_{2} \ln \left (x \right )\right )+\frac {x \ln \left (x \right )^{2}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{2} +\ln \left (x \right ) c_{1} +\frac {\ln \left (x \right )^{2}}{2}\right ) \]

Problem 1110

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=\frac {1}{1+{\mathrm e}^{-x}}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{2 x}-c_{2} {\mathrm e}^{x}+{\mathrm e}^{x} \left (\ln \left (1+{\mathrm e}^{x}\right ) {\mathrm e}^{x}-\ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{x}+\ln \left (1+{\mathrm e}^{-x}\right )-1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (\ln \left (1+{\mathrm e}^{x}\right ) \left (1+{\mathrm e}^{x}\right )+\left (-1-{\mathrm e}^{x}\right ) \ln \left ({\mathrm e}^{x}\right )+{\mathrm e}^{x} c_{1} +c_{2} -1\right ) \]

Problem 1111

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=7 x^{\frac {3}{2}} {\mathrm e}^{x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = {\mathrm e}^{x} \left (c_{2} x +c_{1} \right )+\frac {4 x^{\frac {7}{2}} {\mathrm e}^{x}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{2} +c_{1} x +\frac {4 x^{\frac {7}{2}}}{5}\right ) \]

Problem 1112

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y=4 \sqrt {x}\, {\mathrm e}^{x} \left (4 x +1\right )} \] Given that one solution of the ode is \begin {align*} y_1 &= \sqrt {x}\, {\mathrm e}^{x} \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{x} \left (c_{1} x -c_{2} \right )}{\sqrt {x}}+{\mathrm e}^{x} \sqrt {x}\, \left (\ln \left (x \right )+2 x -1\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x \ln \left (x \right )+2 x^{2}+\left (c_{1} -1\right ) x +c_{2} \right ) {\mathrm e}^{x}}{\sqrt {x}} \]

Problem 1113

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+2 y={\mathrm e}^{x} \sec \left (x \right )} \] Given that one solution of the ode is \begin {align*} y_1 &= \cos \left (x \right ) {\mathrm e}^{x} \end {align*}
program solution	\[ y = {\mathrm e}^{x} \left (c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )\right )+{\mathrm e}^{x} \left (\ln \left (\cos \left (x \right )\right ) \cos \left (x \right )+\sin \left (x \right ) x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-\cos \left (x \right ) \ln \left (\sec \left (x \right )\right )+\cos \left (x \right ) c_{1} +\sin \left (x \right ) \left (c_{2} +x \right )\right ) {\mathrm e}^{x} \]

Problem 1114

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}+2\right ) y=8 \,{\mathrm e}^{-x \left (2+x \right )}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x^{2}} \end {align*}
program solution	\[ y = {\mathrm e}^{-x^{2}} \left (c_{2} x +c_{1} \right )+2 \,{\mathrm e}^{-x \left (2+x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-x^{2}}+2 \,{\mathrm e}^{-x \left (2+x \right )} \]

Problem 1115

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -4 y=-6 x -4} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
program solution	\[ y = c_{1} x^{2}-\frac {c_{2}}{4 x^{2}}+1+2 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2}}{x^{2}}+c_{1} x^{2}+2 x +1 \]

Problem 1116

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 x \left (x -1\right ) y^{\prime }+\left (x^{2}-2 x +2\right ) y={\mathrm e}^{2 x} x^{3}} \] Given that one solution of the ode is \begin {align*} y_1 &= x \,{\mathrm e}^{-x} \end {align*}
program solution	\[ y = {\mathrm e}^{-x} x \left (c_{2} x +c_{1} \right )+\frac {x \,{\mathrm e}^{2 x}}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x \left ({\mathrm e}^{2 x}+9 c_{1} x \,{\mathrm e}^{-x}+9 \,{\mathrm e}^{-x} c_{2} \right )}{9} \]

Problem 1117

ODE	\[ \boxed {x^{2} y^{\prime \prime }-x \left (2 x -1\right ) y^{\prime }+\left (x^{2}-x -1\right ) y=x^{2} {\mathrm e}^{x}} \] Given that one solution of the ode is \begin {align*} y_1 &= x \,{\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x} x -\frac {c_{2} {\mathrm e}^{x}}{2 x}+\frac {x^{2} {\mathrm e}^{x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x} \left (3 c_{1} x^{2}+x^{3}+3 c_{2} \right )}{3 x} \]

Problem 1118

ODE	\[ \boxed {\left (1-2 x \right ) y^{\prime \prime }+2 y^{\prime }+\left (-3+2 x \right ) y=\left (4 x^{2}-4 x +1\right ) {\mathrm e}^{x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x}+x \,{\mathrm e}^{-x} c_{2} -\frac {{\mathrm e}^{x} x \left (x -1\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-x}-\frac {{\mathrm e}^{x} \left (x^{2}-2 c_{2} -x \right )}{2} \]

Problem 1119

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +4 y=4 x^{4}} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
program solution	\[ y = x^{2} \left (c_{1} +c_{2} \ln \left (x \right )\right )+x^{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (\ln \left (x \right ) c_{1} +x^{2}+c_{2} \right ) \]

Problem 1120

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (4 x +1\right ) y^{\prime }+\left (1+2 x \right ) y=3 \sqrt {x}\, {\mathrm e}^{-x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x} \end {align*}
program solution	\[ y = {\mathrm e}^{-x} \left (c_{1} +2 \sqrt {x}\, c_{2} \right )+x^{\frac {3}{2}} {\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (c_{2} +c_{1} \sqrt {x}+x^{\frac {3}{2}}\right ) \]

Problem 1121

ODE	\[ \boxed {x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+\left (x +1\right ) y=-{\mathrm e}^{-x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = {\mathrm e}^{x} \left (c_{1} +\frac {c_{2} x^{2}}{2}\right )-\operatorname {expIntegral}_{1}\left (2 x \right ) {\mathrm e}^{x} x^{2}+\frac {{\mathrm e}^{-x} \left (2 x -1\right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\operatorname {expIntegral}_{1}\left (2 x \right ) x^{2} {\mathrm e}^{x}+\frac {{\mathrm e}^{-x} \left (2 x -1\right )}{4}+{\mathrm e}^{x} \left (c_{1} x^{2}+c_{2} \right ) \]

Problem 1122

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-4 x \left (x +1\right ) y^{\prime }+\left (2 x +3\right ) y=4 x^{\frac {5}{2}} {\mathrm e}^{2 x}} \] Given that one solution of the ode is \begin {align*} y_1 &= \sqrt {x} \end {align*}
program solution	\[ y = \left (c_{2} {\mathrm e}^{x}+c_{1} \right ) \sqrt {x}+\frac {{\mathrm e}^{2 x} \sqrt {x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} +{\mathrm e}^{x} c_{1} +\frac {{\mathrm e}^{2 x}}{2}\right ) \sqrt {x} \]

Problem 1123

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime } x +8 y=4 x^{2}} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
program solution	\[ y = c_{1} x^{2}+\frac {c_{2} x^{4}}{2}+x^{2} \left (-1-2 \ln \left (x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{2} x^{2}-2 \ln \left (x \right )+c_{1} -1\right ) \]

Problem 1124

ODE	\[ \boxed {x y^{\prime \prime }+\left (-2 x +2\right ) y^{\prime }+\left (-2+x \right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x}-\frac {c_{2} {\mathrm e}^{x}}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x} \left (c_{1} x +c_{2} \right )}{x} \]

Problem 1125

ODE	\[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +6 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*}
program solution	\[ y = c_{2} x^{3}+c_{1} x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{1} x +c_{2} \right ) \]

Problem 1126

ODE	\[ \boxed {x^{2} \ln \left (x \right )^{2} y^{\prime \prime }-2 x \ln \left (x \right ) y^{\prime }+\left (2+\ln \left (x \right )\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \ln \left (x \right ) \end {align*}
program solution	\[ y = c_{1} \ln \left (x \right )+\ln \left (x \right ) c_{2} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \ln \left (x \right ) \left (c_{2} x +c_{1} \right ) \]

Problem 1127

ODE	\[ \boxed {4 x y^{\prime \prime }+2 y^{\prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (\sqrt {x}\right ) \end {align*}
program solution	\[ y = \sin \left (\sqrt {x}\right ) c_{1} -2 c_{2} \sin \left (\sqrt {x}\right ) \cot \left (\sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\sqrt {x}\right )+c_{2} \cos \left (\sqrt {x}\right ) \]

Problem 1128

ODE	\[ \boxed {x y^{\prime \prime }-\left (2+2 x \right ) y^{\prime }+\left (2+x \right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x}+\frac {c_{2} {\mathrm e}^{x} x^{3}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{2} x^{3}+c_{1} \right ) \]

Problem 1129

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (2 a -1\right ) x y^{\prime }+a^{2} y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{a} \end {align*}
program solution	\[ y = c_{1} x^{a}+c_{2} x^{a} \ln \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) x^{a} \]

Problem 1130

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (x^{2}+2\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x \right ) x \end {align*}
program solution	\[ y = \sin \left (x \right ) x c_{1} -c_{2} \sin \left (x \right ) \cot \left (x \right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right )\right ) \]

Problem 1131

ODE	\[ \boxed {x y^{\prime \prime }-\left (4 x +1\right ) y^{\prime }+\left (4 x +2\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} x^{2} {\mathrm e}^{2 x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{2} x^{2}+c_{1} \right ) \]

Problem 1132

ODE	\[ \boxed {4 x^{2} \sin \left (x \right ) y^{\prime \prime }-4 x \left (x \cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime }+\left (2 x \cos \left (x \right )+3 \sin \left (x \right )\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sqrt {x} \end {align*}
program solution	\[ y = \sqrt {x}\, c_{1} -c_{2} \sqrt {x}\, \cos \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} +c_{2} \cos \left (x \right )\right ) \]

Problem 1133

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-4 y^{\prime } x +\left (-16 x^{2}+3\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \sqrt {x} \end {align*}
program solution	\[ y = {\mathrm e}^{2 x} \sqrt {x}\, c_{1} -\frac {c_{2} {\mathrm e}^{2 x} \sqrt {x}\, {\mathrm e}^{-4 x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} \sinh \left (2 x \right )+c_{2} \cosh \left (2 x \right )\right ) \]

Problem 1134

ODE	\[ \boxed {\left (1+2 x \right ) x y^{\prime \prime }-2 \left (2 x^{2}-1\right ) y^{\prime }-4 \left (x +1\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{x} \end {align*}
program solution	\[ y = \frac {c_{1}}{x}+c_{2} {\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} {\mathrm e}^{2 x} x +c_{1}}{x} \]

Problem 1135

ODE	\[ \boxed {\left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 x -2\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{x}-c_{2} {\mathrm e}^{x} x^{2} {\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2}+c_{2} {\mathrm e}^{x} \]

Problem 1136

ODE	\[ \boxed {x y^{\prime \prime }-\left (4 x +1\right ) y^{\prime }+\left (4 x +2\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} x^{2} {\mathrm e}^{2 x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{2} x^{2}+c_{1} \right ) \]

Problem 1137

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +4 y=4 x^{4}} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{2} \end {align*} With initial conditions \begin {align*} [y \left (-1\right ) = 7, y^{\prime }\left (-1\right ) = -8] \end {align*}
program solution	\[ y = \left (8 i \pi -8 \ln \left (x \right )+x^{2}+6\right ) x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (8 i \pi +x^{2}-8 \ln \left (x \right )+6\right ) x^{2} \]

Problem 1138

ODE	\[ \boxed {\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*} With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = -x \,{\mathrm e}^{-x}+2 \,{\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{2 x}-x \,{\mathrm e}^{-x} \]

Problem 1139

ODE	\[ \boxed {\left (x +1\right )^{2} y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }-\left (x^{2}+2 x -1\right ) y=\left (x +1\right )^{3} {\mathrm e}^{x}} \] Given that one solution of the ode is \begin {align*} y_1 &= \left (x +1\right ) {\mathrm e}^{x} \end {align*} With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {\left (2 x \,{\mathrm e}^{x}-3 \,{\mathrm e}^{x}+7 \,{\mathrm e}^{-x}\right ) \left (x +1\right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x +1\right ) \left (x \,{\mathrm e}^{x}-5 \sinh \left (x \right )+2 \cosh \left (x \right )\right )}{2} \]

Problem 1140

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -2 y=x^{2}} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*} With initial conditions \begin {align*} \left [y \left (1\right ) = {\frac {5}{4}}, y^{\prime }\left (1\right ) = {\frac {3}{2}}\right ] \end {align*}
program solution	\[ y = \frac {x \left (x +4\right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x +\frac {1}{4} x^{2} \]

Problem 1141

ODE	\[ \boxed {\left (x^{2}-4\right ) y^{\prime \prime }+4 y^{\prime } x +2 y=2+x} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{-2+x} \end {align*} With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {1}{3}}, y^{\prime }\left (0\right ) = -1\right ] \end {align*}
program solution	\[ y = \frac {x^{3}+6 x^{2}+24 x +8}{6 x^{2}-24} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{3}+6 x^{2}+24 x +8}{6 x^{2}-24} \]

Problem 1142

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

Problem 1143

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

Problem 1144

ODE	\[ \boxed {y^{\prime }+y^{2}+5 y=6} \]
program solution	\[ y = \frac {6+{\mathrm e}^{7 x} c_{1}^{7}}{{\mathrm e}^{7 x} c_{1}^{7}-1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {6+c_{1} {\mathrm e}^{7 x}}{c_{1} {\mathrm e}^{7 x}-1} \]

Problem 1145

ODE	\[ \boxed {y^{\prime }+y^{2}+8 y=-7} \]
program solution	\[ y = -\frac {-7+{\mathrm e}^{6 x} c_{1}^{6}}{{\mathrm e}^{6 x} c_{1}^{6}-1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {7-c_{1} {\mathrm e}^{6 x}}{c_{1} {\mathrm e}^{6 x}-1} \]

Problem 1146

ODE	\[ \boxed {y^{\prime }+y^{2}+14 y=-50} \]
program solution	\[ y = -7-\tan \left (x +c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -7-\tan \left (c_{1} +x \right ) \]

Problem 1147

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

Problem 1148

ODE	\[ \boxed {36 y^{\prime }+36 y^{2}-12 y=-1} \]
program solution	\[ y = \frac {c_{1} +x +6}{6 x +6 c_{1}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} +x +6}{6 x +6 c_{1}} \]

Problem 1149

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

Problem 1150

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

Problem 1151

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

Problem 1152

ODE	\[ \boxed {\left (3 x -1\right ) \left (y^{\prime }+y^{2}\right )-y \left (3 x +2\right )=6 x -8} \]
program solution	\[ y = \frac {2 \,{\mathrm e}^{3 x} c_{3} -x +1}{{\mathrm e}^{3 x} c_{3} +x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-c_{1} x +2 \,{\mathrm e}^{3 x -1}+c_{1}}{c_{1} x +{\mathrm e}^{3 x -1}} \]

Problem 1153

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

Problem 1154

ODE	\[ \boxed {x^{2} \left (y^{\prime }+y^{2}\right )-7 y x=-7} \]
program solution	\[ y = \frac {7 x^{6}+c_{3}}{x \left (x^{6}+c_{3} \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-7 x^{6}+c_{1}}{x \left (-x^{6}+c_{1} \right )} \]

Problem 1155

ODE	\[ \boxed {y^{\prime \prime }+9 y=\tan \left (3 x \right )} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}-\frac {\cos \left (3 x \right ) \ln \left (\sec \left (3 x \right )+\tan \left (3 x \right )\right )}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (3 x \right ) c_{2} +\cos \left (3 x \right ) c_{1} -\frac {\cos \left (3 x \right ) \ln \left (\sec \left (3 x \right )+\tan \left (3 x \right )\right )}{9} \]

Problem 1156

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

Problem 1157

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

Problem 1158

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

Problem 1159

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=14 x^{\frac {3}{2}} {\mathrm e}^{x}} \]
program solution	\[ y = {\mathrm e}^{x} \left (c_{2} x +c_{1} \right )+\frac {8 x^{\frac {7}{2}} {\mathrm e}^{x}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{2} +c_{1} x +\frac {8 x^{\frac {7}{2}}}{5}\right ) \]

Problem 1160

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

Problem 1161

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

Problem 1162

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (-2 x +2\right ) y^{\prime }+\left (-2+x \right ) y={\mathrm e}^{2 x}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 1163

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

Problem 1164

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

Problem 1165

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

Problem 1166

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

Problem 1167

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

Problem 1168

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

Problem 1169

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

Problem 1170

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

Problem 1171

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

Problem 1172

ODE	\[ \boxed {x y^{\prime \prime }-y^{\prime }-4 y x^{3}=8 x^{5}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x^{2}}+\frac {c_{2} {\mathrm e}^{x^{2}}}{4}-2 x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sinh \left (x^{2}\right ) c_{2} +\cosh \left (x^{2}\right ) c_{1} -2 x^{2} \]

Problem 1173

ODE	\[ \boxed {\sin \left (x \right ) y^{\prime \prime }+\left (2 \sin \left (x \right )-\cos \left (x \right )\right ) y^{\prime }+\left (-\cos \left (x \right )+\sin \left (x \right )\right ) y={\mathrm e}^{-x}} \]
program solution
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{\arcsin \left (\cos \left (x \right )\right )} \left (\left (\int \csc \left (x \right )^{2} {\mathrm e}^{-\arcsin \left (\cos \left (x \right )\right )-x}d x \right ) \cos \left (x \right )-\cos \left (x \right ) c_{1} -\left (\int \cot \left (x \right ) \csc \left (x \right ) {\mathrm e}^{-\arcsin \left (\cos \left (x \right )\right )-x}d x \right )-c_{2} \right ) \]

Problem 1174

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-4 y^{\prime } x +\left (-16 x^{2}+3\right ) y=8 x^{\frac {5}{2}}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x} \sqrt {x}+\frac {c_{2} {\mathrm e}^{2 x} \sqrt {x}}{4}-\frac {\sqrt {x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (-\frac {1}{2}+\sinh \left (2 x \right ) c_{2} +\cosh \left (2 x \right ) c_{1} \right ) \]

Problem 1175

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-4 y^{\prime } x +\left (4 x^{2}+3\right ) y=x^{\frac {7}{2}}} \]
program solution	\[ y = \sqrt {x}\, \left (c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )\right )+\frac {x^{\frac {3}{2}}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {x}\, \left (x +4 \sin \left (x \right ) c_{2} +4 \cos \left (x \right ) c_{1} \right )}{4} \]

Problem 1176

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

Problem 1177

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

Problem 1178

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x -3 y=x^{\frac {3}{2}}} \]
program solution	\[ y = x^{3} \left (-\frac {4}{15 x^{\frac {3}{2}}}-\frac {c_{1}}{4 x^{4}}+c_{2} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {15 c_{2} x^{4}-4 x^{\frac {5}{2}}+15 c_{1}}{15 x} \]

Problem 1179

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

Problem 1180

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

Problem 1181

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

Problem 1182

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

Problem 1183

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

Problem 1184

ODE	\[ \boxed {\left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y=\left (3 x -1\right )^{2} {\mathrm e}^{2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {\left (3 x^{2}-2 x +6\right ) {\mathrm e}^{2 x}}{6}+\frac {x \,{\mathrm e}^{-x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (3 x^{2}-2 x +6\right ) {\mathrm e}^{2 x}}{6}+\frac {x \,{\mathrm e}^{-x}}{3} \]

Problem 1185

ODE	\[ \boxed {\left (x -1\right )^{2} y^{\prime \prime }-2 \left (x -1\right ) y^{\prime }+2 y=\left (x -1\right )^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -6] \end {align*}
program solution	\[ y = -\left (x -1\right ) \left (\left (1-x \right ) \ln \left (x -1\right )+i x \pi -i \pi -2 x +3\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-i \pi x +i \pi +\ln \left (x -1\right ) x -\ln \left (x -1\right )+2 x -3\right ) \left (x -1\right ) \]

Problem 1186

ODE	\[ \boxed {\left (x -1\right )^{2} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+\left (x -1\right )^{3} y=\left (x -1\right )^{3} {\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = -6] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 1187

ODE	\[ \boxed {\left (x -1\right )^{2} y^{\prime \prime }+4 y^{\prime } x +2 y=2 x} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = -2] \end {align*}
program solution	\[ y = \frac {x^{3}+28 \,\operatorname {expIntegral}_{1}\left (\frac {4}{x -1}\right ) {\mathrm e}^{\frac {4}{x -1}}-28 \,{\mathrm e}^{\frac {4}{x -1}} \operatorname {expIntegral}_{1}\left (-4\right )-7 \,{\mathrm e}^{\frac {4 x}{x -1}}-2 x^{2}-6 x +7}{3 \left (x -1\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{3}-28 \,{\mathrm e}^{\frac {4}{x -1}} \operatorname {expIntegral}_{1}\left (-4\right )-7 \,{\mathrm e}^{\frac {4 x}{x -1}}+28 \,{\mathrm e}^{\frac {4}{x -1}} \operatorname {expIntegral}_{1}\left (\frac {4}{x -1}\right )-2 x^{2}-6 x +7}{3 \left (x -1\right )^{2}} \]

Problem 1188

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

Problem 1189

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

Problem 1190

ODE	\[ \boxed {\left (2+x \right ) y^{\prime \prime }+y^{\prime } x +3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{4} x^{2}+\frac {1}{8} x^{3}+\frac {1}{8} x^{4}-\frac {9}{160} x^{5}+\frac {1}{240} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{40} x^{5}-\frac {13}{720} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{4} x^{2}+\frac {1}{8} x^{3}+\frac {1}{8} x^{4}-\frac {9}{160} x^{5}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{40} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{4} x^{2}+\frac {1}{8} x^{3}+\frac {1}{8} x^{4}-\frac {9}{160} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{40} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1191

ODE	\[ \boxed {\left (3 x^{2}+1\right ) y^{\prime \prime }+3 y^{\prime } x^{2}-2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x^{2}-\frac {1}{3} x^{4}-\frac {3}{10} x^{5}+\frac {17}{45} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {1}{4} x^{4}-\frac {4}{15} x^{5}+\frac {11}{60} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x^{2}-\frac {1}{3} x^{4}-\frac {3}{10} x^{5}\right ) c_{1} +\left (x +\frac {1}{3} x^{3}-\frac {1}{4} x^{4}-\frac {4}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x^{2}-\frac {1}{3} x^{4}-\frac {3}{10} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {1}{4} x^{4}-\frac {4}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1192

ODE	\[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }+\left (-3 x +2\right ) y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-2 x^{2}+\frac {4}{3} x^{3}-\frac {1}{3} x^{4}-\frac {1}{3} x^{5}+\frac {13}{45} x^{6}\right ) y \left (0\right )+\left (x -x^{2}+\frac {1}{2} x^{3}-\frac {1}{12} x^{4}-\frac {17}{120} x^{5}+\frac {11}{120} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-2 x^{2}+\frac {4}{3} x^{3}-\frac {1}{3} x^{4}-\frac {1}{3} x^{5}\right ) c_{1} +\left (x -x^{2}+\frac {1}{2} x^{3}-\frac {1}{12} x^{4}-\frac {17}{120} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-2 x^{2}+\frac {4}{3} x^{3}-\frac {1}{3} x^{4}-\frac {1}{3} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {1}{2} x^{3}-\frac {1}{12} x^{4}-\frac {17}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1193

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+\left (2-x \right ) y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {3}{2} x^{2}+x^{3}-\frac {1}{8} x^{4}-\frac {1}{4} x^{5}+\frac {31}{240} x^{6}\right ) y \left (0\right )+\left (x -x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}-\frac {2}{15} x^{5}+\frac {1}{72} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {3}{2} x^{2}+x^{3}-\frac {1}{8} x^{4}-\frac {1}{4} x^{5}\right ) c_{1} +\left (x -x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}-\frac {2}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {3}{2} x^{2}+x^{3}-\frac {1}{8} x^{4}-\frac {1}{4} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}-\frac {2}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1194

ODE	\[ \boxed {\left (3 x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-2 x^{2}+x^{4}-\frac {16}{15} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {4}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (x^{4}-2 x^{2}+1\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {4}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{4}-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {4}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 1195

ODE	\[ \boxed {x y^{\prime \prime }+\left (4+2 x \right ) y^{\prime }+\left (2+x \right ) y=0} \] With the expansion point for the power series method at \(x = -1\).
program solution	\[ y = \left (1+\frac {\left (x +1\right )^{2}}{2}+\frac {2 \left (x +1\right )^{3}}{3}+\frac {7 \left (x +1\right )^{4}}{8}+\frac {17 \left (x +1\right )^{5}}{15}+\frac {1021 \left (x +1\right )^{6}}{720}\right ) y \left (-1\right )+\left (x +1+\left (x +1\right )^{2}+\frac {3 \left (x +1\right )^{3}}{2}+2 \left (x +1\right )^{4}+\frac {103 \left (x +1\right )^{5}}{40}+\frac {129 \left (x +1\right )^{6}}{40}\right ) y^{\prime }\left (-1\right )+O\left (\left (x +1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {\left (x +1\right )^{2}}{2}+\frac {2 \left (x +1\right )^{3}}{3}+\frac {7 \left (x +1\right )^{4}}{8}+\frac {17 \left (x +1\right )^{5}}{15}\right ) y \left (-1\right )+\left (x +1+\left (x +1\right )^{2}+\frac {3 \left (x +1\right )^{3}}{2}+2 \left (x +1\right )^{4}+\frac {103 \left (x +1\right )^{5}}{40}\right ) D\left (y \right )\left (-1\right )+O\left (x^{6}\right ) \]

Problem 1196

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -3 y x=0} \] With the expansion point for the power series method at \(x = 2\).
program solution	\[ y = \left (1+\frac {3 \left (-2+x \right )^{2}}{4}-\frac {3 \left (-2+x \right )^{3}}{8}+\frac {9 \left (-2+x \right )^{4}}{32}-\frac {27 \left (-2+x \right )^{5}}{160}+\frac {63 \left (-2+x \right )^{6}}{640}\right ) y \left (2\right )+\left (-2+x -\frac {\left (-2+x \right )^{2}}{2}+\frac {\left (-2+x \right )^{3}}{2}-\frac {5 \left (-2+x \right )^{4}}{16}+\frac {31 \left (-2+x \right )^{5}}{160}-\frac {9 \left (-2+x \right )^{6}}{80}\right ) y^{\prime }\left (2\right )+O\left (\left (-2+x \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\frac {3 \left (-2+x \right )^{2}}{4}-\frac {3 \left (-2+x \right )^{3}}{8}+\frac {9 \left (-2+x \right )^{4}}{32}-\frac {27 \left (-2+x \right )^{5}}{160}\right ) y \left (2\right )+\left (-2+x -\frac {\left (-2+x \right )^{2}}{2}+\frac {\left (-2+x \right )^{3}}{2}-\frac {5 \left (-2+x \right )^{4}}{16}+\frac {31 \left (-2+x \right )^{5}}{160}\right ) D\left (y \right )\left (2\right )+O\left (x^{6}\right ) \]

Problem 1197

ODE	\[ \boxed {\left (2-x \right ) y^{\prime \prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = a_{0}, y^{\prime }\left (0\right ) = a_{1}] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x a_{1} +a_{0} -\frac {a_{0} x^{2}}{2}-\frac {x^{3} a_{0}}{12}-\frac {x^{3} a_{1}}{6}+\frac {x^{4} a_{0}}{48}-\frac {x^{4} a_{1}}{24}+\frac {x^{5} a_{0}}{96}-\frac {x^{5} a_{1}}{240}+\frac {a_{0} x^{6}}{360}+O\left (x^{6}\right ) \] Verified OK. \[ y = a_{0} -\frac {a_{0} x^{2}}{2}-\frac {x^{3} a_{0}}{12}+\frac {x^{4} a_{0}}{48}+\frac {x^{5} a_{0}}{96}+x a_{1} -\frac {x^{3} a_{1}}{6}-\frac {x^{4} a_{1}}{24}-\frac {x^{5} a_{1}}{240}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = a_{0} +a_{1} x -\frac {1}{2} a_{0} x^{2}+\left (-\frac {a_{1}}{6}-\frac {a_{0}}{12}\right ) x^{3}+\left (\frac {a_{0}}{48}-\frac {a_{1}}{24}\right ) x^{4}+\left (-\frac {a_{1}}{240}+\frac {a_{0}}{96}\right ) x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 1198

ODE	\[ \boxed {\left (x +1\right ) y^{\prime \prime }+2 \left (x -1\right )^{2} y^{\prime }+3 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = a_{0}, y^{\prime }\left (1\right ) = a_{1}] \end {align*} With the expansion point for the power series method at \(x = 1\).
program solution	\[ y = \left (x -1\right ) a_{1} +a_{0} -\frac {3 a_{0} \left (x -1\right )^{2}}{4}+\frac {\left (x -1\right )^{3} a_{0}}{8}-\frac {\left (x -1\right )^{3} a_{1}}{4}+\frac {\left (x -1\right )^{4} a_{0}}{16}-\frac {\left (x -1\right )^{4} a_{1}}{48}+\frac {3 \left (x -1\right )^{5} a_{0}}{64}+\frac {\left (x -1\right )^{5} a_{1}}{40}-\frac {\left (x -1\right )^{6} a_{0}}{32}+\frac {17 \left (x -1\right )^{6} a_{1}}{960}+O\left (\left (x -1\right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = a_{0} +a_{1} \left (x -1\right )-\frac {3}{4} a_{0} \left (x -1\right )^{2}+\left (\frac {a_{0}}{8}-\frac {a_{1}}{4}\right ) \left (x -1\right )^{3}+\left (\frac {a_{0}}{16}-\frac {a_{1}}{48}\right ) \left (x -1\right )^{4}+\left (\frac {3 a_{0}}{64}+\frac {a_{1}}{40}\right ) \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right ) \]

Problem 1199

ODE	\[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {c_{1} \left (1+2 x +\frac {x^{2}}{3}+O\left (x^{6}\right )\right )}{x}+c_{2} \left (\frac {9 \left (1+2 x +\frac {x^{2}}{3}+O\left (x^{6}\right )\right ) \ln \left (x \right )}{x}+\frac {1-45 x^{2}-16 x^{3}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (x \right ) \left (9 x +18 x^{2}+3 x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +c_{1} \left (1+2 x +\frac {1}{3} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) x +\left (1-5 x -55 x^{2}-\frac {53}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]

Problem 1200

ODE	\[ \boxed {x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (1+2 x \right ) y^{\prime }-\left (6 x +4\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+O\left (x^{6}\right )\right )+c_{2} \left (-4 x^{2} \left (1+O\left (x^{6}\right )\right ) \ln \left (x \right )+\frac {1-\frac {4 x}{3}+2 x^{2}-4 x^{3}+4 x^{5}+O\left (x^{6}\right )}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{2} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (\ln \left (x \right ) \left (576 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+192 x -288 x^{2}+576 x^{3}-576 x^{4}-576 x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]