Systems of ordinary differential equations

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Problem: Find the general solution of the inhomogeneous system of differential equations: \[ \left\{\begin{array}{l} x_{t}^{\prime}=-x+2 y+\frac{3 e^{t}}{t^{2}+4} \\ y_{t}^{\prime}=-2 x+3 y-\frac{2 e^{t}}{t^{2}+4} \end{array}\right. \]

8.3.1 Systems of ordinary differential equations

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Problem: Find the general solution of the inhomogeneous system of differential equations: \[ \left\{\begin{array}{l} x_{t}^{\prime}=3 x-4 y+e^{-2 t} \\ y_{t}^{\prime}=x-2 y-3 e^{-2 t} \end{array}\right. \]

8.3.2 Systems of ordinary differential equations

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Problem: Find the general solution of the inhomogeneous system of differential equations: \[ \left\{\begin{array}{c} x^{\prime}=6 x-2 y-z+5 \\ y^{\prime}=-x+5 y-z+e^{t} \\ z^{\prime}=x-2 y+4 z+2 \end{array}\right. \]

8.3.3 Systems of ordinary differential equations

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Problem: Find the particular solution of the homogeneous system of differential equations: \[ \left\{\begin{array}{ll} \dot{x}=2 x+y, & x(0)=1 \\ \dot{y}=x+2 y, & y(0)=3 \end{array}\right. \]

8.3.4 Systems of ordinary differential equations

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Problem: Find the general solution of the inhomogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=-2 x+3 y+e^{4 t} \\ \dot{y}=2 x-y \end{array}\right. \]

8.3.5 Systems of ordinary differential equations

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Problem: Find the general solution (the first integrals) of the system of differential equations: \[ \frac{d x}{x\left(y^{2}-z^{2}\right)}=\frac{d y}{-y\left(x^{2}+z^{2}\right)}=\frac{d z}{z\left(x^{2}+y^{2}\right)} \text {. } \]

8.3.6 Systems of ordinary differential equations

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Problem: Find the solution of the inhomogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{\bar{x}}=\left(\begin{array}{ll} 0 & 3 \\ 3 & 0 \end{array}\right) \bar{x}+\left(\begin{array}{c} -1 \\ t \end{array}\right) \\ \bar{x}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right) \end{array}\right. \]

8.3.8 Systems of ordinary differential equations

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Problem: Solve the system of homogeneous linear differential equations, using Euler's method: \[ \dot{x}=\left(\begin{array}{ccc} 4 & -5 & 7 \\ 1 & -4 & 9 \\ -4 & 5 & 0 \end{array}\right) \cdot X \text {. } \]

8.3.9 Systems of ordinary differential equations

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Problem: Find the general solution of the homogeneous system of differential equations: \[ \left\{\begin{array}{c} x^{\prime}=7 x+y \\ y^{\prime}=-z \\ z^{\prime}=2 z+y-2 x \end{array}\right. \]

8.3.10 Systems of ordinary differential equations

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Problem: Find the general solution of the homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=2 x-y-z \\ \dot{y}=3 x-2 y-3 z \\ \dot{z}=-x+y+2 z \end{array}\right. \]

8.3.11 Systems of ordinary differential equations

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Problem: Find the general solution of the linear homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=8 x-3 y \\ \dot{y}=2 x+y \end{array}\right. \]

8.3.12 Systems of ordinary differential equations

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Problem: Find the general solution of the linear homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=-y \\ \dot{y}=x \end{array} .\right. \]

8.3.13 Systems of ordinary differential equations

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Problem: Find the general solution of the homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=x+2 y-z \\ \dot{y}=x+y+z \\ \dot{z}=x-z \end{array}\right. \]

8.3.14 Systems of ordinary differential equations

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Problem: Find the particular solution of the system of linear differential equations: \[ \left\{\begin{array}{l} \dot{x}=\frac{x}{x+3 y} \\ \dot{y}=\frac{3 y}{x+3 y} \end{array} \quad x(1)=y(1)=1 .\right. \]

8.3.7 Systems of ordinary differential equations

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Problem: Find the general solution of the linear system: \[ \left\{\begin{array}{c} \dot{x}=4 x+y-e^{2 t} \\ \dot{y}=y-2 x \end{array}\right. \]

8.3.15 Systems of ordinary differential equations

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