Problem 12801
| ODE |
\[ \boxed {y^{\prime \prime }+9 y=18 \,{\mathrm e}^{3 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 6] \end {align*}
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program solution |
\[ y = -2 \cos \left (3 x \right )+\sin \left (3 x \right )+{\mathrm e}^{3 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -2 \cos \left (3 x \right )+\sin \left (3 x \right )+{\mathrm e}^{3 x} \] |
Problem 12802
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 3] \end {align*}
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program solution |
\[ y = -{\mathrm e}^{-x}+{\mathrm e}^{2 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -{\mathrm e}^{-x}+{\mathrm e}^{2 x} \] |
Problem 12803
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=x^{2}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {11}{4}}, y^{\prime }\left (0\right ) = {\frac {1}{2}}\right ] \end {align*}
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program solution |
\[ y = -\frac {3}{4}-\frac {x^{2}}{2}+\frac {7 \,{\mathrm e}^{-x}}{3}+\frac {7 \,{\mathrm e}^{2 x}}{6}+\frac {x}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {7 \,{\mathrm e}^{-x}}{3}+\frac {x}{2}-\frac {x^{2}}{2}+\frac {7 \,{\mathrm e}^{2 x}}{6}-\frac {3}{4} \] |
Problem 12804
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=2 \sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left (3 x -3\right ) {\mathrm e}^{x}+\cos \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (3 x -3\right ) {\mathrm e}^{x}+\cos \left (x \right ) \] |
Problem 12805
| ODE |
\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }+4 y^{\prime }-4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 5, y^{\prime \prime }\left (0\right ) = 5] \end {align*}
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program solution |
\[ y = {\mathrm e}^{x}-\cos \left (2 x \right )+2 \sin \left (2 x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{x}-\cos \left (2 x \right )+2 \sin \left (2 x \right ) \] |
Problem 12806
| ODE |
\[ \boxed {y^{\prime }+2 y=\left \{\begin {array}{cc} 2 & 0\le x <1 \\ 1 & 1\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 1 & x <1 \\ \frac {1}{2}+\frac {{\mathrm e}^{-2 x +2}}{2} & 1\le x \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left \{\begin {array}{cc} 1 & x <1 \\ \frac {1}{2}+\frac {{\mathrm e}^{2-2 x}}{2} & 1\le x \end {array}\right . \] |
Problem 12807
| ODE |
\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=\left \{\begin {array}{cc} 1 & 2\le x <4 \\ 0 & \operatorname {otherwise} \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[
y = \frac {\left (\left \{\begin {array}{cc} -{\mathrm e}^{-x}+{\mathrm e}^{2 x} & x <2 \\ -\frac {1}{2}+{\mathrm e}^{4}-{\mathrm e}^{-2} & x =2 \\ -{\mathrm e}^{-x}+{\mathrm e}^{2 x}+\frac {{\mathrm e}^{2 x -4}}{2}-\frac {3}{2}+{\mathrm e}^{2-x} & x <4 \\ \frac {\left (2 \,{\mathrm e}^{12}+{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{2}-2\right ) {\mathrm e}^{-4}}{2} & x =4 \\ -{\mathrm e}^{-x}+{\mathrm e}^{2 x}-\frac {{\mathrm e}^{2 x -8}}{2}+\frac {{\mathrm e}^{2 x -4}}{2}+{\mathrm e}^{2-x}-{\mathrm e}^{4-x} & 4 |
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Maple solution | \[
y \left (x \right ) = \frac {\left (\left \{\begin {array}{cc} -{\mathrm e}^{-x}+{\mathrm e}^{2 x} & x <2 \\ -\frac {1}{2}-{\mathrm e}^{-2}+{\mathrm e}^{4} & x =2 \\ -{\mathrm e}^{-x}+{\mathrm e}^{2 x}-\frac {3}{2}+{\mathrm e}^{2-x}+\frac {{\mathrm e}^{2 x -4}}{2} & x <4 \\ \frac {\left (2 \,{\mathrm e}^{12}+{\mathrm e}^{8}-2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{2}-2\right ) {\mathrm e}^{-4}}{2} & x =4 \\ -{\mathrm e}^{-x}+{\mathrm e}^{2 x}-{\mathrm e}^{4-x}+{\mathrm e}^{2-x}-\frac {{\mathrm e}^{2 x -8}}{2}+\frac {{\mathrm e}^{2 x -4}}{2} & 4 |
Problem 12808
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }=\left \{\begin {array}{cc} 0 & 0\le x <1 \\ \left (x -1\right )^{2} & 1\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} 1 & x <1 \\ \frac {7}{8} & x =1 \\ \frac {25}{24}+\frac {{\mathrm e}^{2 x -2}}{8}-\frac {x^{3}}{6}+\frac {x^{2}}{4}-\frac {x}{4} & 1 |
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Maple solution | \[
y \left (x \right ) = \left \{\begin {array}{cc} 1 & x <1 \\ \frac {7}{8} & x =1 \\ \frac {25}{24}+\frac {{\mathrm e}^{2 x -2}}{8}+\frac {x^{2}}{4}-\frac {x^{3}}{6}-\frac {x}{4} & 1 |
Problem 12809
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=\left \{\begin {array}{cc} 0 & 0\le x <1 \\ x^{2}-2 x +3 & 1\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} x \,{\mathrm e}^{x} & x <1 \\ {\mathrm e}+8 & x =1 \\ x \,{\mathrm e}^{x}+4 \,{\mathrm e}^{x -1} \left (x -3\right )+x^{2}+2 x +5 & 1 |
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Maple solution | \[
y \left (x \right ) = \left \{\begin {array}{cc} {\mathrm e}^{x} x & x <1 \\ {\mathrm e}+8 & x =1 \\ {\mathrm e}^{x} x +5+4 \left (-3+x \right ) {\mathrm e}^{-1+x}+x^{2}+2 x & 1 |
Problem 12810
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\left \{\begin {array}{cc} 0 & 0\le x <\pi \\ -\sin \left (3 x \right ) & \pi \le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} \cos \left (2 x \right )+\frac {\sin \left (2 x \right )}{2} & x \le \pi \\ \frac {\left (4 \cos \left (x \right )^{2}+8 \cos \left (x \right )-1\right ) \sin \left (x \right )}{5}+2 \cos \left (x \right )^{2}-1 & \pi |
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Maple solution | \[ y \left (x \right ) = \cos \left (2 x \right )+\left (\left \{\begin {array}{cc} \frac {\sin \left (2 x \right )}{2} & x <\pi \\ \frac {4 \sin \left (2 x \right )}{5}+\frac {\sin \left (3 x \right )}{5} & \pi \le x \end {array}\right .\right ) \] |
Problem 12811
| ODE |
\[ \boxed {y^{\prime \prime }-4 y=\left \{\begin {array}{cc} x & 0\le x <1 \\ 1 & 1\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[
y = \frac {\left (\left \{\begin {array}{cc} \sinh \left (2 x \right )-2 x & x <1 \\ \sinh \left (2\right )-4 & x =1 \\ \sinh \left (2 x \right )-\sinh \left (2 x -2\right )-2 & 1 |
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Maple solution | \[
y \left (x \right ) = \frac {\left (\left \{\begin {array}{cc} \sinh \left (2 x \right )-2 x & x <1 \\ \sinh \left (2\right )-4 & x =1 \\ \sinh \left (2 x \right )-\sinh \left (2 x -2\right )-2 & 1 |
Problem 12812
| ODE |
\[ \boxed {y^{\prime \prime }-4 y^{\prime }+5 y=\left \{\begin {array}{cc} x & 0\le x <1 \\ 1 & 1\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[
y = \frac {\left (\left \{\begin {array}{cc} 4+5 x +{\mathrm e}^{2 x} \left (21 \cos \left (x \right )-47 \sin \left (x \right )\right ) & x <1 \\ 10+{\mathrm e}^{2} \left (21 \cos \left (1\right )-47 \sin \left (1\right )\right ) & x =1 \\ \left (4 \cos \left (x -1\right )-3 \sin \left (x -1\right )\right ) {\mathrm e}^{2 x -2}+5+{\mathrm e}^{2 x} \left (21 \cos \left (x \right )-47 \sin \left (x \right )\right ) & 1 |
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Maple solution | \[
y \left (x \right ) = \frac {\left (\left \{\begin {array}{cc} 4+5 x +{\mathrm e}^{2 x} \left (21 \cos \left (x \right )-47 \sin \left (x \right )\right ) & x <1 \\ 10+{\mathrm e}^{2} \left (21 \cos \left (1\right )-47 \sin \left (1\right )\right ) & x =1 \\ \left (4 \cos \left (-1+x \right )-3 \sin \left (-1+x \right )\right ) {\mathrm e}^{2 x -2}+5+{\mathrm e}^{2 x} \left (21 \cos \left (x \right )-47 \sin \left (x \right )\right ) & 1 |
Problem 12813
| ODE |
\[ \boxed {y^{\prime }+3 y=\delta \left (x -2\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} {\mathrm e}^{-3 x} & x <2 \\ {\mathrm e}^{-3 x}+{\mathrm e}^{-3 x +6} & 2\le x \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \operatorname {Heaviside}\left (x -2\right ) {\mathrm e}^{6-3 x}+{\mathrm e}^{-3 x} \] |
Problem 12814
| ODE |
\[ \boxed {-3 y+y^{\prime }=\delta \left (x -1\right )+2 \operatorname {Heaviside}\left (x -2\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} 0 & x \le 1 \\ {\mathrm e}^{3 x -3} & x <2 \\ -\frac {2}{3}+{\mathrm e}^{3} & x =2 \\ -\frac {2}{3}+\frac {2 \,{\mathrm e}^{3 x -6}}{3}+{\mathrm e}^{3 x -3} & 2 |
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Maple solution | \[ y \left (x \right ) = -\frac {2 \operatorname {Heaviside}\left (x -2\right )}{3}+\frac {2 \operatorname {Heaviside}\left (x -2\right ) {\mathrm e}^{-6+3 x}}{3}+\operatorname {Heaviside}\left (-1+x \right ) {\mathrm e}^{3 x -3} \] |
Problem 12815
| ODE |
\[ \boxed {y^{\prime \prime }+9 y=\delta \left (x -\pi \right )+\delta \left (x -3 \pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = -\frac {\sin \left (3 x \right ) \left (\left \{\begin {array}{cc} 0 & x <\pi \\ 1 & x <3 \pi \\ 2 & 3 \pi \le x \end {array}\right .\right )}{3} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (\operatorname {Heaviside}\left (x -3 \pi \right )+\operatorname {Heaviside}\left (x -\pi \right )\right ) \sin \left (3 x \right )}{3} \] |
Problem 12816
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=2 \delta \left (x -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[
y = x \,{\mathrm e}^{x}+\left (\left \{\begin {array}{cc} 0 & x \le 1 \\ 2 \,{\mathrm e}^{x -1} \left (x -1\right ) & 1 |
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Maple solution | \[ y \left (x \right ) = 2 \operatorname {Heaviside}\left (-1+x \right ) {\mathrm e}^{-1+x} \left (-1+x \right )+{\mathrm e}^{x} x \] |
Problem 12817
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=\cos \left (x \right )+2 \delta \left (x -\pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[
y = \frac {4 \,{\mathrm e}^{x} \cos \left (2 x \right )}{5}+\frac {\cos \left (x \right )}{5}-\frac {\sin \left (x \right )}{10}-\frac {7 \sin \left (2 x \right ) {\mathrm e}^{x}}{20}+\left (\left \{\begin {array}{cc} 0 & x \le \pi \\ \sin \left (2 x \right ) {\mathrm e}^{x -\pi } & \pi |
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Maple solution | \[ y \left (x \right ) = \sin \left (2 x \right ) \operatorname {Heaviside}\left (x -\pi \right ) {\mathrm e}^{x -\pi }+\frac {4 \,{\mathrm e}^{x} \cos \left (2 x \right )}{5}-\frac {7 \,{\mathrm e}^{x} \sin \left (2 x \right )}{20}-\frac {\sin \left (x \right )}{10}+\frac {\cos \left (x \right )}{5} \] |
Problem 12818
| ODE |
\[ \boxed {y^{\prime \prime }+4 y=\cos \left (x \right ) \delta \left (x -\pi \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[
y = \left \{\begin {array}{cc} \frac {\sin \left (2 x \right )}{2} & x \le \pi \\ 0 & \pi |
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Maple solution | \[ y \left (x \right ) = -\frac {\sin \left (2 x \right ) \left (-1+\operatorname {Heaviside}\left (x -\pi \right )\right )}{2} \] |
Problem 12819
| ODE |
\[ \boxed {y^{\prime \prime }+a^{2} y=\delta \left (x -\pi \right ) f \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \left \{\begin {array}{cc} 0 & x <\pi \\ \frac {f \left (\pi \right ) \sin \left (a \left (x -\pi \right )\right )}{a} & \pi \le x \end {array}\right . \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\operatorname {Heaviside}\left (x -\pi \right ) \sin \left (a \left (x -\pi \right )\right ) f \left (\pi \right )}{a} \] |
Problem 12820
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )-2 y_{2} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{-x} \\ y_{2} \left (x \right ) &= \frac {c_{1} {\mathrm e}^{x}}{3}+c_{2} {\mathrm e}^{-x} \\ \end{align*} |
Problem 12821
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{1} \left (x \right )-2 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+3 y_{2} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{2 x} \left (\sin \left (x \right ) c_{1} +\cos \left (x \right ) c_{2} \right ) \\ y_{2} \left (x \right ) &= -\frac {{\mathrm e}^{2 x} \left (\sin \left (x \right ) c_{1} -\sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{1} +\cos \left (x \right ) c_{2} \right )}{2} \\ \end{align*} |
Problem 12822
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{1} \left (x \right )+2 y_{2} \left (x \right )+x -1\\ y_{2}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )+2 y_{2} \left (x \right )-5 x -2 \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = -2, y_{2} \left (0\right ) = 3] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= -2+3 x \\ y_{2} \left (x \right ) &= 3-2 x \\ \end{align*} |
Problem 12823
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=\frac {2 y_{1} \left (x \right )}{x}-\frac {y_{2} \left (x \right )}{x^{2}}-3+\frac {1}{x}-\frac {1}{x^{2}}\\ y_{2}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )+1-6 x \end {align*} With initial conditions \[ [y_{1} \left (1\right ) = -2, y_{2} \left (1\right ) = -5] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= -2 x \\ y_{2} \left (x \right ) &= -1+x \left (-5 x +1\right ) \\ \end{align*} |
Problem 12824
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=\frac {5 y_{1} \left (x \right )}{x}+\frac {4 y_{2} \left (x \right )}{x}-2 x\\ y_{2}^{\prime }\left (x \right )&=-\frac {6 y_{1} \left (x \right )}{x}-\frac {5 y_{2} \left (x \right )}{x}+5 x \end {align*} With initial conditions \[ [y_{1} \left (-1\right ) = 3, y_{2} \left (-1\right ) = -3] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= \frac {2 x^{3}+x^{2}-2}{x} \\ y_{2} \left (x \right ) &= -\frac {2 x^{3}+2 x^{2}-6}{2 x} \\ \end{align*} |
Problem 12825
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )-2 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-y_{1} \left (x \right )+y_{2} \left (x \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 1, y_{2} \left (0\right ) = -1] \] |
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= \left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x}+\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x} \\ y_{2} \left (x \right ) &= -\frac {\left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x} \sqrt {3}}{2}+\frac {\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x} \sqrt {3}}{2}+\frac {\left (\frac {1}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (2+\sqrt {3}\right ) x}}{2}+\frac {\left (\frac {1}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{-\left (-2+\sqrt {3}\right ) x}}{2} \\ \end{align*} |
Problem 12826
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=\sin \left (x \right ) y_{1} \left (x \right )+\sqrt {x}\, y_{2} \left (x \right )+\ln \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=\tan \left (x \right ) y_{1} \left (x \right )-{\mathrm e}^{x} y_{2} \left (x \right )+1 \end {align*} With initial conditions \[ [y_{1} \left (1\right ) = 1, y_{2} \left (1\right ) = -1] \] |
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 12827
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=\sin \left (x \right ) y_{1} \left (x \right )+\sqrt {x}\, y_{2} \left (x \right )+\ln \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=\tan \left (x \right ) y_{1} \left (x \right )-{\mathrm e}^{x} y_{2} \left (x \right )+1 \end {align*} With initial conditions \[ [y_{1} \left (2\right ) = 1, y_{2} \left (2\right ) = -1] \] |
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 12828
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&={\mathrm e}^{-x} y_{1} \left (x \right )-\sqrt {1+x}\, y_{2} \left (x \right )+x^{2}\\ y_{2}^{\prime }\left (x \right )&=\frac {y_{1} \left (x \right )}{x^{2}-4 x +4} \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 0, y_{2} \left (0\right ) = 1] \] |
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 12829
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&={\mathrm e}^{-x} y_{1} \left (x \right )-\sqrt {1+x}\, y_{2} \left (x \right )+x^{2}\\ y_{2}^{\prime }\left (x \right )&=\frac {y_{1} \left (x \right )}{x^{2}-4 x +4} \end {align*} With initial conditions \[ [y_{1} \left (3\right ) = 1, y_{2} \left (3\right ) = 0] \] |
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 12838
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )+5 \,{\mathrm e}^{x}\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+4 y_{2} \left (x \right )-2 \,{\mathrm e}^{-x} \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{3 x} \cos \left (\sqrt {2}\, x \right ) c_{2} +{\mathrm e}^{3 x} \sin \left (\sqrt {2}\, x \right ) c_{1} +\frac {{\mathrm e}^{-x}}{3}-\frac {5 \,{\mathrm e}^{x}}{2} \\ y_{2} \left (x \right ) &= -\frac {{\mathrm e}^{3 x} \cos \left (\sqrt {2}\, x \right ) c_{2}}{3}+\frac {{\mathrm e}^{3 x} \sqrt {2}\, \sin \left (\sqrt {2}\, x \right ) c_{2}}{3}-\frac {{\mathrm e}^{3 x} \sin \left (\sqrt {2}\, x \right ) c_{1}}{3}-\frac {{\mathrm e}^{3 x} \sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) c_{1}}{3}+\frac {{\mathrm e}^{-x}}{3}+\frac {5 \,{\mathrm e}^{x}}{6} \\ \end{align*} |
Problem 12839
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{2} \left (x \right )-2 y_{1} \left (x \right )+2 \sin \left (x \right ) \cos \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-3 y_{1} \left (x \right )+y_{2} \left (x \right )-8 \cos \left (x \right )^{3}+6 \cos \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1} +\frac {16 \cos \left (3 x \right )}{73}-\frac {4 \cos \left (2 x \right )}{13}-\frac {6 \sin \left (3 x \right )}{73}+\frac {7 \sin \left (2 x \right )}{13} \\ y_{2} \left (x \right ) &= \frac {3 c_{2} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {c_{2} {\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {3 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1}}{2}-\frac {{\mathrm e}^{-\frac {x}{2}} \sqrt {3}\, \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1}}{2}-\frac {60 \sin \left (3 x \right )}{73}+\frac {9 \sin \left (2 x \right )}{13}+\frac {14 \cos \left (3 x \right )}{73}+\frac {6 \cos \left (2 x \right )}{13} \\ \end{align*} |
Problem 12840
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=2 y_{3} \left (x \right )-y_{1} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{\sqrt {6}\, x} c_{2} +c_{3} {\mathrm e}^{-\sqrt {6}\, x} \\ y_{2} \left (x \right ) &= \frac {\sqrt {6}\, \left ({\mathrm e}^{\sqrt {6}\, x} c_{2} -c_{3} {\mathrm e}^{-\sqrt {6}\, x}\right )}{2} \\ y_{3} \left (x \right ) &= \frac {2 \,{\mathrm e}^{2 x} c_{1}}{\left (2+\sqrt {6}\right ) \left (-2+\sqrt {6}\right )}+\frac {{\mathrm e}^{-\sqrt {6}\, x} c_{3}}{2+\sqrt {6}}-\frac {{\mathrm e}^{\sqrt {6}\, x} c_{2}}{-2+\sqrt {6}} \\ \end{align*} |
Problem 12841
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right ) x -x^{2} y_{2} \left (x \right )+4 x\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right ) {\mathrm e}^{x}+3 \,{\mathrm e}^{-x} y_{2} \left (x \right )-4 \cos \left (x \right )^{3}+3 \cos \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\[ \text {No solution found} \] |
Problem 12842
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ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )-2 y_{2} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{-x} \\ y_{2} \left (x \right ) &= \frac {c_{1} {\mathrm e}^{x}}{3}+c_{2} {\mathrm e}^{-x} \\ \end{align*} |
Problem 12843
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )+4 x -2\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )-2 y_{2} \left (x \right )+3 x \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{2} {\mathrm e}^{x}+{\mathrm e}^{-x} c_{1} +x \\ y_{2} \left (x \right ) &= \frac {c_{2} {\mathrm e}^{x}}{3}+{\mathrm e}^{-x} c_{1} -1+2 x \\ \end{align*} |
Problem 12844
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=\frac {5 y_{1} \left (x \right )}{x}+\frac {4 y_{2} \left (x \right )}{x}\\ y_{2}^{\prime }\left (x \right )&=-\frac {6 y_{1} \left (x \right )}{x}-\frac {5 y_{2} \left (x \right )}{x} \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= \frac {c_{1} x^{2}+c_{2}}{x} \\ y_{2} \left (x \right ) &= -\frac {2 c_{1} x^{2}+3 c_{2}}{2 x} \\ \end{align*} |
Problem 12845
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=\frac {5 y_{1} \left (x \right )}{x}+\frac {4 y_{2} \left (x \right )}{x}-2 x\\ y_{2}^{\prime }\left (x \right )&=-\frac {6 y_{1} \left (x \right )}{x}-\frac {5 y_{2} \left (x \right )}{x}+5 x \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= \frac {c_{1} x^{2}+2 x^{3}+c_{2}}{x} \\ y_{2} \left (x \right ) &= -\frac {2 c_{1} x^{2}+2 x^{3}+3 c_{2}}{2 x} \\ \end{align*} |
Problem 12846
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )+y_{2} \left (x \right )-2 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=3 y_{2} \left (x \right )-2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )+y_{2} \left (x \right )-3 y_{3} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{1} {\mathrm e}^{x}+\frac {c_{2} {\mathrm e}^{2 x}}{2}+c_{3} {\mathrm e}^{-x} \\ y_{2} \left (x \right ) &= c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{2 x}+c_{3} {\mathrm e}^{-x} \\ y_{3} \left (x \right ) &= c_{1} {\mathrm e}^{x}+\frac {c_{2} {\mathrm e}^{2 x}}{2}+2 c_{3} {\mathrm e}^{-x} \\ \end{align*} |
Problem 12847
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=5 y_{1} \left (x \right )-5 y_{2} \left (x \right )-5 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-y_{1} \left (x \right )+4 y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )-5 y_{2} \left (x \right )-3 y_{3} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{2 x} \left (\sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{3} \right ) \\ y_{2} \left (x \right ) &= -\frac {\left (2 \sin \left (x \right ) c_{2} -\sin \left (x \right ) c_{3} +\cos \left (x \right ) c_{2} +2 \cos \left (x \right ) c_{3} -5 c_{1} \right ) {\mathrm e}^{2 x}}{5} \\ y_{3} \left (x \right ) &= {\mathrm e}^{2 x} \left (\sin \left (x \right ) c_{2} +\cos \left (x \right ) c_{3} -c_{1} \right ) \\ \end{align*} |
Problem 12848
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=4 y_{1} \left (x \right )+6 y_{2} \left (x \right )+6 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+3 y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=-y_{1} \left (x \right )-4 y_{2} \left (x \right )-3 y_{3} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{2} {\mathrm e}^{4 x}+c_{3} {\mathrm e}^{-x} \\ y_{2} \left (x \right ) &= \frac {c_{2} {\mathrm e}^{4 x}}{3}+\frac {c_{3} {\mathrm e}^{-x}}{3}+c_{1} {\mathrm e}^{x} \\ y_{3} \left (x \right ) &= -\frac {7 c_{3} {\mathrm e}^{-x}}{6}-\frac {c_{2} {\mathrm e}^{4 x}}{3}-c_{1} {\mathrm e}^{x} \\ \end{align*} |
Problem 12849
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{1} \left (x \right )+2 y_{2} \left (x \right )-3 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-3 y_{1} \left (x \right )+4 y_{2} \left (x \right )-2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )+y_{3} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= \frac {{\mathrm e}^{2 x} \left (3 \cos \left (3 x \right ) c_{2} +\cos \left (3 x \right ) c_{3} +\sin \left (3 x \right ) c_{2} -3 \sin \left (3 x \right ) c_{3} +c_{1} \right )}{2} \\ y_{2} \left (x \right ) &= \frac {{\mathrm e}^{2 x} \left (6 \cos \left (3 x \right ) c_{2} -2 \cos \left (3 x \right ) c_{3} -2 \sin \left (3 x \right ) c_{2} -6 \sin \left (3 x \right ) c_{3} +7 c_{1} \right )}{4} \\ y_{3} \left (x \right ) &= {\mathrm e}^{2 x} \left (c_{1} +\sin \left (3 x \right ) c_{2} +\cos \left (3 x \right ) c_{3} \right ) \\ \end{align*} |
Problem 12850
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=-2 y_{1} \left (x \right )-y_{2} \left (x \right )+y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-y_{1} \left (x \right )-2 y_{2} \left (x \right )-y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=y_{1} \left (x \right )-y_{2} \left (x \right )-2 y_{3} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{2} +c_{3} {\mathrm e}^{-3 x} \\ y_{2} \left (x \right ) &= -c_{2} -c_{3} {\mathrm e}^{-3 x}+c_{1} {\mathrm e}^{-3 x} \\ y_{3} \left (x \right ) &= -2 c_{3} {\mathrm e}^{-3 x}+c_{2} +c_{1} {\mathrm e}^{-3 x} \\ \end{align*} |
Problem 12851
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{1} \left (x \right )+y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )+2 y_{2} \left (x \right )+4 y_{3} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{2} +c_{3} {\mathrm e}^{6 x} \\ y_{2} \left (x \right ) &= c_{2} +c_{3} {\mathrm e}^{6 x}+c_{1} \\ y_{3} \left (x \right ) &= 2 c_{3} {\mathrm e}^{6 x}-c_{2} -\frac {c_{1}}{2} \\ \end{align*} |
Problem 12852
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )+y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-y_{1} \left (x \right )+2 y_{2} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=3 y_{3} \left (x \right )-4 y_{4} \left (x \right )\\ y_{4}^{\prime }\left (x \right )&=4 y_{3} \left (x \right )+3 y_{4} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{2 x} \left (\sin \left (x \right ) c_{3} +c_{4} \cos \left (x \right )\right ) \\ y_{2} \left (x \right ) &= -{\mathrm e}^{2 x} \left (\sin \left (x \right ) c_{4} -\cos \left (x \right ) c_{3} \right ) \\ y_{3} \left (x \right ) &= {\mathrm e}^{3 x} \left (\cos \left (4 x \right ) c_{2} +\sin \left (4 x \right ) c_{1} \right ) \\ y_{4} \left (x \right ) &= -{\mathrm e}^{3 x} \left (\cos \left (4 x \right ) c_{1} -\sin \left (4 x \right ) c_{2} \right ) \\ \end{align*} |
Problem 12853
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-3 y_{1} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=y_{4} \left (x \right )\\ y_{4}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )-5 y_{3} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= -\frac {c_{1} \left (4+\sqrt {5}\right )^{\frac {3}{2}} \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{11}-\frac {c_{2} \left (4-\sqrt {5}\right )^{\frac {3}{2}} \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{11}-\frac {c_{3} \left (4+\sqrt {5}\right )^{\frac {3}{2}} \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{11}-\frac {c_{4} \left (4-\sqrt {5}\right )^{\frac {3}{2}} \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{11}+\frac {8 c_{1} \sqrt {4+\sqrt {5}}\, \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{11}+\frac {8 c_{2} \sqrt {4-\sqrt {5}}\, \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{11}+\frac {8 c_{3} \sqrt {4+\sqrt {5}}\, \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{11}+\frac {8 c_{4} \sqrt {4-\sqrt {5}}\, \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{11} \\ y_{2} \left (x \right ) &= -c_{1} \sin \left (\sqrt {4+\sqrt {5}}\, x \right )-c_{2} \sin \left (\sqrt {4-\sqrt {5}}\, x \right )+c_{3} \cos \left (\sqrt {4+\sqrt {5}}\, x \right )+c_{4} \cos \left (\sqrt {4-\sqrt {5}}\, x \right ) \\ y_{3} \left (x \right ) &= \frac {13 c_{1} \sqrt {4+\sqrt {5}}\, \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{22}+\frac {13 c_{2} \sqrt {4-\sqrt {5}}\, \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{22}+\frac {13 c_{3} \sqrt {4+\sqrt {5}}\, \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{22}+\frac {13 c_{4} \sqrt {4-\sqrt {5}}\, \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{22}-\frac {3 c_{1} \left (4+\sqrt {5}\right )^{\frac {3}{2}} \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{22}-\frac {3 c_{2} \left (4-\sqrt {5}\right )^{\frac {3}{2}} \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{22}-\frac {3 c_{3} \left (4+\sqrt {5}\right )^{\frac {3}{2}} \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{22}-\frac {3 c_{4} \left (4-\sqrt {5}\right )^{\frac {3}{2}} \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{22} \\ y_{4} \left (x \right ) &= \frac {c_{1} \sin \left (\sqrt {4+\sqrt {5}}\, x \right ) \sqrt {5}}{2}-\frac {c_{2} \sin \left (\sqrt {4-\sqrt {5}}\, x \right ) \sqrt {5}}{2}-\frac {c_{3} \cos \left (\sqrt {4+\sqrt {5}}\, x \right ) \sqrt {5}}{2}+\frac {c_{4} \cos \left (\sqrt {4-\sqrt {5}}\, x \right ) \sqrt {5}}{2}+\frac {c_{1} \sin \left (\sqrt {4+\sqrt {5}}\, x \right )}{2}+\frac {c_{2} \sin \left (\sqrt {4-\sqrt {5}}\, x \right )}{2}-\frac {c_{3} \cos \left (\sqrt {4+\sqrt {5}}\, x \right )}{2}-\frac {c_{4} \cos \left (\sqrt {4-\sqrt {5}}\, x \right )}{2} \\ \end{align*} |
Problem 12854
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )+2 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=-2 y_{1} \left (x \right )+3 y_{2} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=y_{3} \left (x \right )\\ y_{4}^{\prime }\left (x \right )&=2 y_{4} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{3 x} \left (\sin \left (2 x \right ) c_{1} +\cos \left (2 x \right ) c_{2} \right ) \\ y_{2} \left (x \right ) &= -{\mathrm e}^{3 x} \left (\sin \left (2 x \right ) c_{2} -\cos \left (2 x \right ) c_{1} \right ) \\ y_{3} \left (x \right ) &= c_{4} {\mathrm e}^{x} \\ y_{4} \left (x \right ) &= c_{3} {\mathrm e}^{2 x} \\ \end{align*} |
Problem 12855
|
ODE |
\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{2} \left (x \right )+y_{4} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )-y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=y_{4} \left (x \right )\\ y_{4}^{\prime }\left (x \right )&=y_{3} \left (x \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} y_{1} \left (x \right ) &= c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{-x} \\ y_{2} \left (x \right ) &= c_{1} {\mathrm e}^{x}-c_{2} {\mathrm e}^{-x}-c_{3} {\mathrm e}^{x}+c_{4} {\mathrm e}^{-x} \\ y_{3} \left (x \right ) &= c_{3} {\mathrm e}^{x}+c_{4} {\mathrm e}^{-x} \\ y_{4} \left (x \right ) &= c_{3} {\mathrm e}^{x}-c_{4} {\mathrm e}^{-x} \\ \end{align*} |
Problem 12856
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t} \\ y \left (t \right ) &= c_{1} {\mathrm e}^{t}+\frac {c_{2} {\mathrm e}^{-t}}{3} \\ \end{align*} |
Problem 12857
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )+3 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{t} \left (2 c_{2} t +2 c_{1} +c_{2} \right )}{2} \\ \end{align*} |
Problem 12858
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-3 y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-2 t} \left (\sin \left (\sqrt {3}\, t \right ) c_{1} +\cos \left (\sqrt {3}\, t \right ) c_{2} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-2 t} \left (\sqrt {3}\, \sin \left (\sqrt {3}\, t \right ) c_{2} -\sqrt {3}\, \cos \left (\sqrt {3}\, t \right ) c_{1} +\sin \left (\sqrt {3}\, t \right ) c_{1} +\cos \left (\sqrt {3}\, t \right ) c_{2} \right )}{2} \\ \end{align*} |
Problem 12859
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=5 x \left (t \right )+y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right ) \\ y \left (t \right ) &= -\frac {3 c_{1} \cos \left (3 t \right )}{2}+\frac {3 c_{2} \sin \left (3 t \right )}{2}-\frac {c_{1} \sin \left (3 t \right )}{2}-\frac {c_{2} \cos \left (3 t \right )}{2} \\ \end{align*} |
Problem 12860
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )-y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{-t} \left (c_{1} \cos \left (2 t \right )-c_{2} \sin \left (2 t \right )\right ) \\ \end{align*} |
Problem 12861
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )+y \left (t \right ) \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= -{\mathrm e}^{t} \left (c_{1} \cos \left (2 t \right )-c_{2} \sin \left (2 t \right )\right ) \\ \end{align*} |
Problem 12862
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=-5 x \left (t \right )-y \left (t \right )+2\\ y^{\prime }\left (t \right )&=3 x \left (t \right )-y \left (t \right )-3 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= \frac {5}{8}-\frac {{\mathrm e}^{-4 t} c_{1}}{2}+c_{2} {\mathrm e}^{-2 t} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-4 t} c_{1}}{2}-3 c_{2} {\mathrm e}^{-2 t}-\frac {9}{8} \\ \end{align*} |
Problem 12863
|
ODE |
\begin {align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-2 y \left (t \right )-6\\ y^{\prime }\left (t \right )&=4 x \left (t \right )-y \left (t \right )+2 \end {align*}
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program solution |
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Maple solution |
\begin{align*} x \left (t \right ) &= -2+{\mathrm e}^{t} \left (c_{1} \cos \left (2 t \right )+c_{2} \sin \left (2 t \right )\right ) \\ y \left (t \right ) &= -6+{\mathrm e}^{t} \left (c_{1} \cos \left (2 t \right )-c_{2} \cos \left (2 t \right )+c_{1} \sin \left (2 t \right )+c_{2} \sin \left (2 t \right )\right ) \\ \end{align*} |
Problem 12864
| ODE |
\[ \boxed {y^{\prime }-\frac {y+1}{1+t}=0} \] |
|
program solution |
\[ y = {\mathrm e}^{c_{1}} t +{\mathrm e}^{c_{1}}-1 \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = c_{1} t +c_{1} -1 \] |
Problem 12865
| ODE |
\[ \boxed {y^{\prime }-y^{2} t^{2}=0} \] |
|
program solution |
\[ y = -\frac {3}{t^{3}+c_{3}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {3}{t^{3}-3 c_{1}} \] |
Problem 12866
| ODE |
\[ \boxed {y^{\prime }-t^{4} y=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\frac {t^{5}}{5}+c_{1}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = c_{1} {\mathrm e}^{\frac {t^{5}}{5}} \] |
Problem 12867
| ODE |
\[ \boxed {y^{\prime }-2 y=1} \] |
|
program solution |
\[ y = {\mathrm e}^{2 t} c_{1}^{2}-\frac {1}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {1}{2}+c_{1} {\mathrm e}^{2 t} \] |
Problem 12868
| ODE |
\[ \boxed {y^{\prime }+y=2} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{-t}}{c_{1}}+2 \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 2+{\mathrm e}^{-t} c_{1} \] |
Problem 12869
| ODE |
\[ \boxed {y^{\prime }-{\mathrm e}^{-y}=0} \] |
|
program solution |
\[ y = \ln \left (t +c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \ln \left (t +c_{1} \right ) \] |
Problem 12870
| ODE |
\[ \boxed {x^{\prime }-x^{2}=1} \] |
|
program solution |
\[ x = \tan \left (t +c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \tan \left (t +c_{1} \right ) \] |
Problem 12871
| ODE |
\[ \boxed {y^{\prime }-2 t y^{2}-3 y^{2}=0} \] |
|
program solution |
\[ y = -\frac {4}{4 t^{2}+4 c_{3} +12 t +9} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1}{-t^{2}+c_{1} -3 t} \] |
Problem 12872
| ODE |
\[ \boxed {y^{\prime }-\frac {t}{y}=0} \] |
|
program solution |
\[ -\frac {t^{2}}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \sqrt {t^{2}+c_{1}} \\ y \left (t \right ) &= -\sqrt {t^{2}+c_{1}} \\ \end{align*} |
Problem 12873
| ODE |
\[ \boxed {y^{\prime }-\frac {t}{t^{2} y+y}=0} \] |
|
program solution |
\[ -\frac {\ln \left (t^{2}+1\right )}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \sqrt {\ln \left (t^{2}+1\right )+c_{1}} \\ y \left (t \right ) &= -\sqrt {\ln \left (t^{2}+1\right )+c_{1}} \\ \end{align*} |
Problem 12874
| ODE |
\[ \boxed {y^{\prime }-t y^{\frac {1}{3}}=0} \] |
|
program solution |
\[ y = \frac {\left (3 t^{2}+6 c_{1} \right )^{\frac {3}{2}}}{27} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right )^{\frac {2}{3}}-\frac {t^{2}}{3}-c_{1} = 0 \] |
Problem 12875
| ODE |
\[ \boxed {y^{\prime }-\frac {1}{2 y+1}=0} \] |
|
program solution |
\[ y = -\frac {1}{2}-\frac {\sqrt {1+4 t +4 c_{1}}}{2} \] Verified OK. \[ y = -\frac {1}{2}+\frac {\sqrt {1+4 t +4 c_{1}}}{2} \] Verified OK.
|
|
Maple solution |
\begin{align*} y \left (t \right ) &= -\frac {1}{2}-\frac {\sqrt {1+4 c_{1} +4 t}}{2} \\ y \left (t \right ) &= -\frac {1}{2}+\frac {\sqrt {1+4 c_{1} +4 t}}{2} \\ \end{align*} |
Problem 12876
| ODE |
\[ \boxed {y^{\prime }-\frac {2 y+1}{t}=0} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{2 c_{1}} t^{2}}{2}-\frac {1}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {1}{2}+t^{2} c_{1} \] |
Problem 12877
| ODE |
\[ \boxed {y^{\prime }-y \left (-y+1\right )=0} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{t} c_{1}}{-1+c_{1} {\mathrm e}^{t}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1}{1+{\mathrm e}^{-t} c_{1}} \] |
Problem 12878
| ODE |
\[ \boxed {y^{\prime }-\frac {4 t}{1+3 y^{2}}=0} \] |
|
program solution |
\[ -\frac {t^{2}}{2}+\frac {y}{4}+\frac {y^{3}}{4} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \frac {\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 t^{2} c_{1} +324 c_{1}^{2}+3}\right )^{\frac {2}{3}}-3}{3 \left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 t^{2} c_{1} +324 c_{1}^{2}+3}\right )^{\frac {1}{3}}} \\ y \left (t \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 t^{2} c_{1} +324 c_{1}^{2}+3}\right )^{\frac {2}{3}}+3 i \sqrt {3}-3}{6 \left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 t^{2} c_{1} +324 c_{1}^{2}+3}\right )^{\frac {1}{3}}} \\ y \left (t \right ) &= \frac {i \left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 t^{2} c_{1} +324 c_{1}^{2}+3}\right )^{\frac {2}{3}} \sqrt {3}-\left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 t^{2} c_{1} +324 c_{1}^{2}+3}\right )^{\frac {2}{3}}+3 i \sqrt {3}+3}{6 \left (27 t^{2}+54 c_{1} +3 \sqrt {81 t^{4}+324 t^{2} c_{1} +324 c_{1}^{2}+3}\right )^{\frac {1}{3}}} \\ \end{align*} |
Problem 12879
| ODE |
\[ \boxed {v^{\prime }-t^{2} v+2 v=t^{2}-2} \] |
|
program solution |
\[ v = {\mathrm e}^{\frac {1}{3} t^{3}-2 t +c_{1}}-1 \] Verified OK.
|
|
Maple solution | \[ v \left (t \right ) = -1+{\mathrm e}^{\frac {t \left (t^{2}-6\right )}{3}} c_{1} \] |
Problem 12880
| ODE |
\[ \boxed {y^{\prime }-\frac {1}{1+y t +y+t}=0} \] |
|
program solution |
\[ \frac {y^{2}}{2}+y-\ln \left (1+t \right ) = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= -1-\sqrt {1+2 \ln \left (t +1\right )+2 c_{1}} \\ y \left (t \right ) &= -1+\sqrt {1+2 \ln \left (t +1\right )+2 c_{1}} \\ \end{align*} |
Problem 12881
| ODE |
\[ \boxed {y^{\prime }-\frac {{\mathrm e}^{t} y}{1+y^{2}}=0} \] |
|
program solution |
\[ y = {\mathrm e}^{-\frac {\operatorname {LambertW}\left ({\mathrm e}^{2 c_{1} +2 \,{\mathrm e}^{t}}\right )}{2}+c_{1} +{\mathrm e}^{t}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{{\mathrm e}^{t}+c_{1}}}{\sqrt {\frac {{\mathrm e}^{2 c_{1} +2 \,{\mathrm e}^{t}}}{\operatorname {LambertW}\left ({\mathrm e}^{2 c_{1} +2 \,{\mathrm e}^{t}}\right )}}} \] |
Problem 12882
| ODE |
\[ \boxed {y^{\prime }-y^{2}=-4} \] |
|
program solution |
\[ y = -2 \tanh \left (2 t +2 c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {-2 c_{1} {\mathrm e}^{4 t}-2}{-1+c_{1} {\mathrm e}^{4 t}} \] |
Problem 12883
| ODE |
\[ \boxed {w^{\prime }-\frac {w}{t}=0} \] |
|
program solution |
\[ w = {\mathrm e}^{c_{1}} t \] Verified OK.
|
|
Maple solution | \[ w \left (t \right ) = c_{1} t \] |
Problem 12884
| ODE |
\[ \boxed {y^{\prime }-\sec \left (y\right )=0} \] |
|
program solution |
\[ y = \arcsin \left (x +c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (x \right ) = \arcsin \left (c_{1} +x \right ) \] |
Problem 12885
| ODE |
\[ \boxed {x^{\prime }+x t=0} \] With initial conditions \begin {align*} \left [x \left (0\right ) = \frac {1}{\sqrt {\pi }}\right ] \end {align*}
|
|
program solution |
\[ x = \frac {{\mathrm e}^{-\frac {t^{2}}{2}}}{\sqrt {\pi }} \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = \frac {{\mathrm e}^{-\frac {t^{2}}{2}}}{\sqrt {\pi }} \] |
Problem 12886
| ODE |
\[ \boxed {y^{\prime }-y t=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
|
|
program solution |
\[ y = 3 \,{\mathrm e}^{\frac {t^{2}}{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 3 \,{\mathrm e}^{\frac {t^{2}}{2}} \] |
Problem 12887
| ODE |
\[ \boxed {y^{\prime }+y^{2}=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {1}{2}}\right ] \end {align*}
|
|
program solution |
\[ \frac {1}{y} = 2+t \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1}{t +2} \] |
Problem 12888
| ODE |
\[ \boxed {y^{\prime }-t^{2} y^{3}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
|
|
program solution |
\[ -\frac {t^{3}}{3}-\frac {1}{2 y^{2}} = -{\frac {1}{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {3}{\sqrt {-6 t^{3}+9}} \] |
Problem 12889
| ODE |
\[ \boxed {y^{\prime }+y^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = 0 \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 0 \] |
Problem 12890
| ODE |
\[ \boxed {y^{\prime }-\frac {t}{y-t^{2} y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
|
|
program solution |
\[ -\frac {\ln \left (t -1\right )}{2}-\frac {\ln \left (1+t \right )}{2}-\frac {y^{2}}{2} = -\frac {i \pi }{2}-8 \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \sqrt {i \pi -\ln \left (t -1\right )-\ln \left (t +1\right )+16} \] |
Problem 12891
| ODE |
\[ \boxed {y^{\prime }-2 y=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
|
|
program solution |
\[ -\frac {\ln \left (2\right )}{2}+\frac {\ln \left (2 y+1\right )}{2} = t +\frac {\ln \left (7\right )}{2}-\frac {\ln \left (2\right )}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {1}{2}+\frac {7 \,{\mathrm e}^{2 t}}{2} \] |
Problem 12892
| ODE |
\[ \boxed {y^{\prime }-t y^{2}-2 y^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ y = -\frac {2}{t^{2}+4 t -2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {2}{t^{2}+4 t -2} \] |
Problem 12893
| ODE |
\[ \boxed {x^{\prime }-\frac {t^{2}}{x+t^{3} x}=0} \] With initial conditions \begin {align*} [x \left (0\right ) = -2] \end {align*}
|
|
program solution |
\[ -\frac {\ln \left (t^{3}+1\right )}{3}+\frac {x^{2}}{2} = 2 \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -\frac {\sqrt {36+6 \ln \left (t^{3}+1\right )}}{3} \] |
Problem 12894
| ODE |
\[ \boxed {y^{\prime }-\frac {1-y^{2}}{y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2] \end {align*}
|
|
program solution |
\[ -\frac {\ln \left (y^{2}-1\right )}{2} = t -\frac {\ln \left (3\right )}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\sqrt {3 \,{\mathrm e}^{-2 t}+1} \] |
Problem 12895
| ODE |
\[ \boxed {y^{\prime }-\left (1+y^{2}\right ) t=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ y = \frac {-\sin \left (\frac {t^{2}}{2}\right )-\cos \left (\frac {t^{2}}{2}\right )}{\sin \left (\frac {t^{2}}{2}\right )-\cos \left (\frac {t^{2}}{2}\right )} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \tan \left (\frac {t^{2}}{2}+\frac {\pi }{4}\right ) \] |
Problem 12896
| ODE |
\[ \boxed {y^{\prime }-\frac {1}{2 y+3}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
\[ y \left (y+3\right ) = 4+t \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {3}{2}+\frac {\sqrt {25+4 t}}{2} \] |
Problem 12897
| ODE |
\[ \boxed {y^{\prime }-2 t y^{2}-3 y^{2} t^{2}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
|
|
program solution |
\[ y = -\frac {1}{t^{3}+t^{2}-1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {1}{t^{3}+t^{2}-1} \] |
Problem 12898
| ODE |
\[ \boxed {y^{\prime }-\frac {y^{2}+5}{y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2] \end {align*}
|
|
program solution |
\[ \frac {\ln \left (y^{2}+5\right )}{2} = t +\ln \left (3\right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\sqrt {9 \,{\mathrm e}^{2 t}-5} \] |
Problem 12899
| ODE |
\[ \boxed {y^{\prime }=t^{2}+t} \] |
|
program solution |
\[ y = \frac {t^{2} \left (2 t +3\right )}{6}+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1}{3} t^{3}+\frac {1}{2} t^{2}+c_{1} \] |
Problem 12900
| ODE |
\[ \boxed {y^{\prime }=t^{2}+1} \] |
|
program solution |
\[ y = \frac {1}{3} t^{3}+t +c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1}{3} t^{3}+t +c_{1} \] |