2.17.29 Problems 2801 to 2900

Problem 2801

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

Problem 2802

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

Problem 2803

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

Problem 2804

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +9 y=9 \ln \left (x \right )} \]
program solution	\[ y = x^{-3 i} c_{1} -\frac {i c_{2} x^{3 i}}{6}+\left (i x^{3 i}+x^{-3 i}\right ) \operatorname {undefined} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (3 \ln \left (x \right )\right ) c_{2} +\cos \left (3 \ln \left (x \right )\right ) c_{1} +\ln \left (x \right ) \]

Problem 2805

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +5 y=8 x \ln \left (x \right )^{2}} \]
program solution	\[ y = x^{1-2 i} c_{1} -\frac {i c_{2} x^{1+2 i}}{4}+\left (i x^{1+2 i}+x^{1-2 i}\right ) \operatorname {undefined} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (-1+\sin \left (2 \ln \left (x \right )\right ) c_{2} +\cos \left (2 \ln \left (x \right )\right ) c_{1} +2 \ln \left (x \right )^{2}\right ) \]

Problem 2806

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

Problem 2807

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

Problem 2808

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

Problem 2809

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

Problem 2810

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

Problem 2811

ODE	\[ \boxed {t^{2} y^{\prime \prime }+y^{\prime } t +25 y=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = \frac {3 \sqrt {3}}{2}, y^{\prime }\left (1\right ) = {\frac {15}{2}}\right ] \end {align*}
program solution	\[ y = \frac {\left (3 i+3 \sqrt {3}\right ) t^{-5 i}}{4}-\frac {3 \left (i-\sqrt {3}\right ) t^{5 i}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {3 \sin \left (5 \ln \left (t \right )\right )}{2}+\frac {3 \sqrt {3}\, \cos \left (5 \ln \left (t \right )\right )}{2} \]

Problem 2812

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

Problem 2813

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

Problem 2814

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	\[ y = c_{1} x \sin \left (x \right )-c_{2} \sin \left (x \right ) x \cot \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right )\right ) \]

Problem 2815

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

Problem 2816

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

Problem 2817

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

Problem 2818

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

Problem 2819

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

Problem 2820

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

Problem 2821

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=15 \,{\mathrm e}^{3 x} \sqrt {x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{3 x} \end {align*}
program solution	\[ y = {\mathrm e}^{3 x} \left (c_{2} x +c_{1} \right )+4 x^{\frac {5}{2}} {\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (c_{2} +c_{1} x +4 x^{\frac {5}{2}}\right ) \]

Problem 2822

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

Problem 2823

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

Problem 2824

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

Problem 2825

ODE	\[ \boxed {y^{\prime \prime \prime }+11 y^{\prime \prime }+36 y^{\prime }+26 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{\left (-5-i\right ) x} c_{2} +{\mathrm e}^{\left (-5+i\right ) x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{-5 x} \sin \left (x \right )+c_{3} {\mathrm e}^{-5 x} \cos \left (x \right ) \]

Problem 2826

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=4 \,{\mathrm e}^{-3 x}} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (c_{2} x +c_{1} \right )+2 x^{2} {\mathrm e}^{-3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-3 x} \left (c_{1} x +2 x^{2}+c_{2} \right ) \]

Problem 2827

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=4 \,{\mathrm e}^{-2 x}} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (c_{2} x +c_{1} \right )+4 \,{\mathrm e}^{-2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-3 x}+4 \,{\mathrm e}^{-2 x} \]

Problem 2828

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+25 y^{\prime }=x^{2}} \]
program solution	\[ y = c_{1} +{\mathrm e}^{\left (3+4 i\right ) x} c_{2} +{\mathrm e}^{\left (3-4 i\right ) x} c_{3} +\frac {x^{3}}{75}+\frac {6 x^{2}}{625}+\frac {22 x}{15625} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (3 c_{1} -4 c_{2} \right ) \cos \left (4 x \right )+4 \sin \left (4 x \right ) \left (c_{1} +\frac {3 c_{2}}{4}\right )\right ) {\mathrm e}^{3 x}}{25}+\frac {x^{3}}{75}+\frac {6 x^{2}}{625}+\frac {22 x}{15625}+c_{3} \]

Problem 2829

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+25 y^{\prime }=\sin \left (4 x \right )} \]
program solution	\[ y = c_{1} +{\mathrm e}^{\left (3+4 i\right ) x} c_{2} +{\mathrm e}^{\left (3-4 i\right ) x} c_{3} -\frac {\cos \left (4 x \right )}{292}+\frac {2 \sin \left (4 x \right )}{219} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (3 c_{1} -4 c_{2} \right ) \cos \left (4 x \right )+4 \sin \left (4 x \right ) \left (c_{1} +\frac {3 c_{2}}{4}\right )\right ) {\mathrm e}^{3 x}}{25}+c_{3} -\frac {\cos \left (4 x \right )}{292}+\frac {2 \sin \left (4 x \right )}{219} \]

Problem 2830

ODE	\[ \boxed {y^{\prime \prime \prime }+9 y^{\prime \prime }+24 y^{\prime }+16 y=8 \,{\mathrm e}^{-x}+1} \]
program solution	\[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{-4 x}+c_{1} {\mathrm e}^{-x}+\frac {1}{16}+\frac {8 x \,{\mathrm e}^{-x}}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{16}+\frac {\left (-16+24 x +27 c_{2} \right ) {\mathrm e}^{-x}}{27}+\left (c_{3} x +c_{1} \right ) {\mathrm e}^{-4 x} \]

Problem 2831

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

Problem 2832

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

Problem 2833

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

Problem 2834

ODE	\[ \boxed {y^{\prime \prime }+y x=\sin \left (x \right )} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (\frac {1}{3}, \frac {2 x^{\frac {3}{2}}}{3}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (\frac {1}{3}, \frac {2 x^{\frac {3}{2}}}{3}\right )+\pi \left (\left (\int _{0}^{x}\operatorname {AiryBi}\left (-\alpha \right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {AiryAi}\left (-x \right )-\left (\int _{0}^{x}\operatorname {AiryAi}\left (-\alpha \right ) \sin \left (\alpha \right )d \alpha \right ) \operatorname {AiryBi}\left (-x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \pi \left (\int \operatorname {AiryBi}\left (-x \right ) \sin \left (x \right )d x \right ) \operatorname {AiryAi}\left (-x \right )-\pi \left (\int \operatorname {AiryAi}\left (-x \right ) \sin \left (x \right )d x \right ) \operatorname {AiryBi}\left (-x \right )+\operatorname {AiryBi}\left (-x \right ) c_{1} +\operatorname {AiryAi}\left (-x \right ) c_{2} \]

Problem 2835

ODE	\[ \boxed {y^{\prime \prime }+4 y=\ln \left (x \right )} \]
program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {\ln \left (x \right ) \cos \left (2 x \right )^{2}}{4}+\frac {\left (2 \mcoloneq \gamma +2 \ln \left (2\right )-2 \,\operatorname {Ci}\left (2 x \right )\right ) \cos \left (2 x \right )}{8}+\frac {\sin \left (2 x \right ) \left (2 \sin \left (2 x \right ) \ln \left (x \right )+\pi -2 \,\operatorname {Si}\left (2 x \right )\right )}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {i \cos \left (2 x \right ) \pi \left (-1+\operatorname {csgn}\left (x \right )\right ) \operatorname {csgn}\left (i x \right )}{8}+\frac {\left (8 c_{1} -2 \,\operatorname {Ci}\left (2 x \right )\right ) \cos \left (2 x \right )}{8}+\frac {\left (\pi \,\operatorname {csgn}\left (x \right )+8 c_{2} -2 \,\operatorname {Si}\left (2 x \right )\right ) \sin \left (2 x \right )}{8}+\frac {\ln \left (x \right )}{4} \]

Problem 2836

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

Problem 2837

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

Problem 2838

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

Problem 2839

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

Problem 2840

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

Problem 2841

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

Problem 2842

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

Problem 2843

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

Problem 2844

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

Problem 2845

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

Problem 2846

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

Problem 2847

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

Problem 2848

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

Problem 2849

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-12 y=36} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 12] \end {align*}
program solution	\[ y = 3 \,{\mathrm e}^{4 t}-3 \] Verified OK.
Maple solution	\[ y \left (t \right ) = 3 \,{\mathrm e}^{4 t}-3 \]

Problem 2850

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=10 \,{\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = 3 \,{\mathrm e}^{-2 t}-3 \cosh \left (t \right )+7 \sinh \left (t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -3 \cosh \left (t \right )+7 \sinh \left (t \right )+3 \,{\mathrm e}^{-2 t} \]

Problem 2851

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

Problem 2852

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }=30 \,{\mathrm e}^{-3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left (3 \,{\mathrm e}^{5 t}-4 \,{\mathrm e}^{3 t}+2\right ) {\mathrm e}^{-3 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (3 \,{\mathrm e}^{5 t}-4 \,{\mathrm e}^{3 t}+2\right ) {\mathrm e}^{-3 t} \]

Problem 2853

ODE	\[ \boxed {y^{\prime \prime }-y=12 \,{\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = -3 \cosh \left (t \right )-7 \sinh \left (t \right )+4 \,{\mathrm e}^{2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -3 \cosh \left (t \right )-7 \sinh \left (t \right )+4 \,{\mathrm e}^{2 t} \]

Problem 2854

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

Problem 2855

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-6 y=12-6 \,{\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 5, y^{\prime }\left (0\right ) = -3] \end {align*}
program solution	\[ y = \frac {\left (8 \,{\mathrm e}^{5 t}+5 \,{\mathrm e}^{3 t}-10 \,{\mathrm e}^{2 t}+22\right ) {\mathrm e}^{-2 t}}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (8 \,{\mathrm e}^{5 t}+5 \,{\mathrm e}^{3 t}-10 \,{\mathrm e}^{2 t}+22\right ) {\mathrm e}^{-2 t}}{5} \]

Problem 2856

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

Problem 2857

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

Problem 2858

ODE	\[ \boxed {y^{\prime \prime }-y=8 \sin \left (t \right )-6 \cos \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = -4 \sin \left (t \right )+3 \cos \left (t \right )+3 \sinh \left (t \right )-\cosh \left (t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -4 \sin \left (t \right )+3 \cos \left (t \right )+3 \sinh \left (t \right )-\cosh \left (t \right ) \]

Problem 2859

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

Problem 2860

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+4 y=20 \sin \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{-t}-2 \cos \left (2 t \right )-{\mathrm e}^{-4 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 \,{\mathrm e}^{-t}-{\mathrm e}^{-4 t}-2 \cos \left (2 t \right ) \]

Problem 2861

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+4 y=20 \sin \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -2] \end {align*}
program solution	\[ y = \frac {10 \,{\mathrm e}^{-t}}{3}-2 \cos \left (2 t \right )-\frac {{\mathrm e}^{-4 t}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {10 \,{\mathrm e}^{-t}}{3}-\frac {{\mathrm e}^{-4 t}}{3}-2 \cos \left (2 t \right ) \]

Problem 2862

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=3 \cos \left (t \right )+\sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {7 \,{\mathrm e}^{2 t}}{5}+\frac {3 \cos \left (t \right )}{5}-\frac {4 \sin \left (t \right )}{5}-{\mathrm e}^{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {7 \,{\mathrm e}^{2 t}}{5}-{\mathrm e}^{t}-\frac {4 \sin \left (t \right )}{5}+\frac {3 \cos \left (t \right )}{5} \]

Problem 2863

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

Problem 2864

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

Problem 2865

ODE	\[ \boxed {y^{\prime \prime }+9 y=7 \sin \left (4 t \right )+14 \cos \left (4 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = -2 \cos \left (4 t \right )-\sin \left (4 t \right )+3 \cos \left (3 t \right )+2 \sin \left (3 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -2 \cos \left (4 t \right )-\sin \left (4 t \right )+3 \cos \left (3 t \right )+2 \sin \left (3 t \right ) \]

Problem 2866

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

Problem 2867

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

Problem 2868

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

Problem 2869

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

Problem 2870

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

Problem 2871

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

Problem 2872

ODE	\[ \boxed {y^{\prime }-3 y=\left \{\begin {array}{cc} \sin \left (t \right ) & 0\le t <\frac {\pi }{2} \\ 1 & \frac {\pi }{2}\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} \frac {21 \,{\mathrm e}^{3 t}}{10}-\frac {\cos \left (t \right )}{10}-\frac {3 \sin \left (t \right )}{10} & t <\frac {\pi }{2} \\ \frac {21 \,{\mathrm e}^{\frac {3 \pi }{2}}}{10}-\frac {19}{30} & t =\frac {\pi }{2} \\ \frac {21 \,{\mathrm e}^{3 t}}{10}+\frac {{\mathrm e}^{-\frac {3 \pi }{2}+3 t}}{30}-\frac {1}{3} & \frac {\pi }{2}
Maple solution	\[ y \left (t \right ) = \frac {\left (\left \{\begin {array}{cc} 21 \,{\mathrm e}^{3 t}-\cos \left (t \right )-3 \sin \left (t \right ) & t <\frac {\pi }{2} \\ -\frac {19}{3}+21 \,{\mathrm e}^{\frac {3 \pi }{2}} & t =\frac {\pi }{2} \\ 21 \,{\mathrm e}^{3 t}+\frac {{\mathrm e}^{3 t -\frac {3 \pi }{2}}}{3}-\frac {10}{3} & \frac {\pi }{2}

Problem 2873

ODE	\[ \boxed {y^{\prime }-3 y=-10 \,{\mathrm e}^{-t +a} \sin \left (-2 t +2 a \right ) \operatorname {Heaviside}\left (t -a \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 5] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 5 \,{\mathrm e}^{3 t}+\operatorname {Heaviside}\left (-a \right ) \left (-{\mathrm e}^{-3 a +3 t}+{\mathrm e}^{a +3 t} \left (\cos \left (2 a \right )-2 \sin \left (2 a \right )\right )\right ) & t
Maple solution	\[ y \left (t \right ) = \left (\operatorname {Heaviside}\left (t -a \right )+\operatorname {Heaviside}\left (a \right )-1\right ) {\mathrm e}^{-3 a +3 t}-\left (\left (\cos \left (2 t \right )+2 \sin \left (2 t \right )\right ) \cos \left (2 a \right )-2 \sin \left (2 a \right ) \left (\cos \left (2 t \right )-\frac {\sin \left (2 t \right )}{2}\right )\right ) {\mathrm e}^{-t +a} \operatorname {Heaviside}\left (t -a \right )-\left (\operatorname {Heaviside}\left (a \right )-1\right ) \left (\cos \left (2 a \right )-2 \sin \left (2 a \right )\right ) {\mathrm e}^{3 t +a}+5 \,{\mathrm e}^{3 t} \]

Problem 2874

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

Problem 2875

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

Problem 2876

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

Problem 2877

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

Problem 2878

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=-10 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (t +\frac {\pi }{4}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-t}-\left (\left \{\begin {array}{cc} 0 & t <\frac {\pi }{4} \\ 2 \,{\mathrm e}^{-2 t +\frac {\pi }{2}}-5 \,{\mathrm e}^{-t +\frac {\pi }{4}}+\sqrt {2}\, \left (\sin \left (t \right )+2 \cos \left (t \right )\right ) & \frac {\pi }{4}\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -2 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) {\mathrm e}^{\frac {\pi }{2}-2 t}+5 \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) {\mathrm e}^{-t +\frac {\pi }{4}}-2 \sqrt {2}\, \left (\cos \left (t \right )+\frac {\sin \left (t \right )}{2}\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right )-{\mathrm e}^{-2 t}+2 \,{\mathrm e}^{-t} \]

Problem 2879

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=30 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -4] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} \left ({\mathrm e}^{5 t}+2\right ) {\mathrm e}^{-3 t} & t <1 \\ \left ({\mathrm e}^{5}-2 \,{\mathrm e}^{3}+2\right ) {\mathrm e}^{-3} & t =1 \\ \left ({\mathrm e}^{5 t}-5 \,{\mathrm e}^{2 t +1}+2 \,{\mathrm e}^{5 t -2}+3 \,{\mathrm e}^{3}+2\right ) {\mathrm e}^{-3 t} & 1
Maple solution	\[ y \left (t \right ) = \left (-5 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{1+2 t}+3 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{3}+2 \,{\mathrm e}^{-2+5 t} \operatorname {Heaviside}\left (t -1\right )+{\mathrm e}^{5 t}+2\right ) {\mathrm e}^{-3 t} \]

Problem 2880

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+5 y=5 \operatorname {Heaviside}\left (t -3\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = {\mathrm e}^{-2 t} \left (2 \cos \left (t \right )+5 \sin \left (t \right )\right )-\left (\left \{\begin {array}{cc} 0 & t <3 \\ -1+\left (\frac {1}{2}+i\right ) {\mathrm e}^{\left (-2-i\right ) \left (t -3\right )}+\left (\frac {1}{2}-i\right ) {\mathrm e}^{\left (-2+i\right ) \left (t -3\right )} & 3\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-\frac {1}{2}-i\right ) \operatorname {Heaviside}\left (-3+t \right ) {\mathrm e}^{\left (-2-i\right ) \left (-3+t \right )}+\left (-\frac {1}{2}+i\right ) \operatorname {Heaviside}\left (-3+t \right ) {\mathrm e}^{\left (-2+i\right ) \left (-3+t \right )}+\operatorname {Heaviside}\left (-3+t \right )+{\mathrm e}^{-2 t} \left (2 \cos \left (t \right )+5 \sin \left (t \right )\right ) \]

Problem 2881

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=2 \sin \left (t \right )+\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (1+\cos \left (t \right )\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} \frac {\left (-2 \cos \left (t \right )^{2}-\sin \left (t \right ) \cos \left (t \right )+1\right ) {\mathrm e}^{t}}{5}+\frac {\cos \left (t \right )}{5}+\frac {2 \sin \left (t \right )}{5} & t <\frac {\pi }{2} \\ \frac {1}{2}+\frac {{\mathrm e}^{\frac {\pi }{2}}}{5} & t =\frac {\pi }{2} \\ \frac {1}{5}+\frac {\left (2 \cos \left (t \right )^{2}-3 \sin \left (t \right ) \cos \left (t \right )-1\right ) {\mathrm e}^{t -\frac {\pi }{2}}}{10}-\frac {2 \,{\mathrm e}^{t} \cos \left (t \right )^{2}}{5}+\frac {\left (-{\mathrm e}^{t} \sin \left (t \right )+2\right ) \cos \left (t \right )}{5}+\frac {{\mathrm e}^{t}}{5}+\frac {3 \sin \left (t \right )}{10} & \frac {\pi }{2}
Maple solution	\[ y \left (t \right ) = \frac {\left (\left (2 \cos \left (t \right )^{2}-3 \cos \left (t \right ) \sin \left (t \right )-1\right ) {\mathrm e}^{t -\frac {\pi }{2}}+2 \cos \left (t \right )-\sin \left (t \right )+2\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )}{10}-\frac {2 \,{\mathrm e}^{t} \cos \left (t \right )^{2}}{5}-\frac {\sin \left (t \right ) \cos \left (t \right ) {\mathrm e}^{t}}{5}+\frac {\cos \left (t \right )}{5}+\frac {{\mathrm e}^{t}}{5}+\frac {2 \sin \left (t \right )}{5} \]

Problem 2882

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

Problem 2883

ODE	\[ \boxed {y^{\prime }-y=\left \{\begin {array}{cc} 2 & 0\le t <1 \\ -1 & 1\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	N/A
Maple solution	\[ y \left (t \right ) = \left \{\begin {array}{cc} {\mathrm e}^{t} & t <0 \\ -2+3 \,{\mathrm e}^{t} & t <1 \\ 1+3 \,{\mathrm e}^{t}-3 \,{\mathrm e}^{t -1} & 1\le t \end {array}\right . \]

Problem 2884

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

Problem 2885

ODE	\[ \boxed {y^{\prime }-2 y=\delta \left (t -2\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} {\mathrm e}^{2 t} & t \le 2 \\ {\mathrm e}^{2 t}+{\mathrm e}^{2 t -4} & 2
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{-4+2 t}+{\mathrm e}^{2 t} \]

Problem 2886

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

Problem 2887

ODE	\[ \boxed {y^{\prime }-5 y=2 \,{\mathrm e}^{-t}+\delta \left (t -3\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} \frac {2 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{3} & t \le 3 \\ \frac {2 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{3}+{\mathrm e}^{-15+5 t} & 3
Maple solution	\[ y \left (t \right ) = \frac {2 \,{\mathrm e}^{2 t} \sinh \left (3 t \right )}{3}+\operatorname {Heaviside}\left (-3+t \right ) {\mathrm e}^{5 t -15} \]

Problem 2888

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

Problem 2889

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

Problem 2890

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

Problem 2891

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+13 y=\delta \left (t -\frac {\pi }{4}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} {\mathrm e}^{2 t} \left (3 \cos \left (3 t \right )-2 \sin \left (3 t \right )\right ) & t \le \frac {\pi }{4} \\ -\frac {{\mathrm e}^{2 t -\frac {\pi }{2}} \sqrt {2}\, \left (\sin \left (3 t \right )+\cos \left (3 t \right )\right )}{6}+3 \,{\mathrm e}^{2 t} \left (\cos \left (3 t \right )-\frac {2 \sin \left (3 t \right )}{3}\right ) & \frac {\pi }{4}
Maple solution	\[ y \left (t \right ) = -\frac {\sqrt {2}\, {\mathrm e}^{-\frac {\pi }{2}+2 t} \operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \left (\sin \left (3 t \right )+\cos \left (3 t \right )\right )}{6}+3 \,{\mathrm e}^{2 t} \left (\cos \left (3 t \right )-\frac {2 \sin \left (3 t \right )}{3}\right ) \]

Problem 2892

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

Problem 2893

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+13 y=\delta \left (t -\frac {\pi }{4}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 5, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = 5 \,{\mathrm e}^{-3 t} \left (\cos \left (2 t \right )+2 \sin \left (2 t \right )\right )-\left (\left \{\begin {array}{cc} 0 & t <\frac {\pi }{4} \\ \frac {{\mathrm e}^{-3 t +\frac {3 \pi }{4}} \cos \left (2 t \right )}{2} & \frac {\pi }{4}\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -\frac {\pi }{4}\right ) \cos \left (2 t \right ) {\mathrm e}^{\frac {3 \pi }{4}-3 t}}{2}+5 \,{\mathrm e}^{-3 t} \left (\cos \left (2 t \right )+2 \sin \left (2 t \right )\right ) \]

Problem 2894

ODE	\[ \boxed {y^{\prime \prime }+9 y=15 \sin \left (2 t \right )+\delta \left (t -\frac {\pi }{6}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = 3 \sin \left (2 t \right )-2 \sin \left (3 t \right )-\left (\left \{\begin {array}{cc} 0 & t <\frac {\pi }{6} \\ \frac {\cos \left (3 t \right )}{3} & \frac {\pi }{6}\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\cos \left (3 t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{6}\right )}{3}-2 \sin \left (3 t \right )+3 \sin \left (2 t \right ) \]

Problem 2895

ODE	\[ \boxed {y^{\prime \prime }+16 y=4 \cos \left (3 t \right )+\delta \left (t -\frac {\pi }{3}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {4 \cos \left (4 t \right )}{7}+\frac {4 \cos \left (3 t \right )}{7}+\left (\left \{\begin {array}{cc} 0 & t <\frac {\pi }{3} \\ \frac {\cos \left (4 t \right ) \sqrt {3}}{8}-\frac {\sin \left (4 t \right )}{8} & \frac {\pi }{3}\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\cos \left (4 t \right ) \sqrt {3}-\sin \left (4 t \right )\right ) \operatorname {Heaviside}\left (t -\frac {\pi }{3}\right )}{8}-\frac {4 \cos \left (4 t \right )}{7}+\frac {4 \cos \left (3 t \right )}{7} \]

Problem 2896

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=4 \sin \left (t \right )+\delta \left (t -\frac {\pi }{6}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = -\left (\left \{\begin {array}{cc} 0 & t <\frac {\pi }{6} \\ \frac {{\mathrm e}^{-t +\frac {\pi }{6}} \left (\cos \left (2 t \right ) \sqrt {3}-\sin \left (2 t \right )\right )}{4} & \frac {\pi }{6}\le t \end {array}\right .\right )+\frac {{\mathrm e}^{-t} \left (4 \cos \left (2 t \right )+3 \sin \left (2 t \right )\right )}{10}-\frac {2 \cos \left (t \right )}{5}+\frac {4 \sin \left (t \right )}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -\frac {\pi }{6}\right ) \left (\sqrt {3}\, \cos \left (t \right )^{2}-\cos \left (t \right ) \sin \left (t \right )-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-t +\frac {\pi }{6}}}{2}+\frac {\left (4 \cos \left (t \right )^{2}+3 \cos \left (t \right ) \sin \left (t \right )-2\right ) {\mathrm e}^{-t}}{5}-\frac {2 \cos \left (t \right )}{5}+\frac {4 \sin \left (t \right )}{5} \]

Problem 2897

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

Problem 2898

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

Problem 2899

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

Problem 2900

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