Higher order differential equations

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Problem: Investigate linear dependence of the vectors \( e^{x}, e^{2 x}, e^{3 x} \) in \( (-\infty,+\infty) \).

8.1.3.28 Higher order differential equations

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Problem: Find the homogeneous linear differential equation with constant real coefficients, possibly of a lower order, a) which has the solution \( y_{1}=e^{4 x} \sin 2 x \), b) which has the solution \( y_{1}=x e^{-4 x} \sin 2 x \), c) which has the solutions \( y_{1}=x^{3}, y_{2}=e^{4 x} \sin 2 x \). clip2net.com

8.1.3.9 Higher order differential equations

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Problem: Find the general solution of the linear homogeneous differential equation with constant coefficients: \[ y^{\prime \prime \prime}+5 y^{\prime \prime}-6 y^{\prime}=0 . \] clip2net.com

8.1.3.13 Higher order differential equations

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Problem: Find the general solution of the linear homogeneous differential equation with constants coefficients: \[ y^{\prime \prime \prime}-4 y^{\prime \prime}+5 y^{\prime}-2 y=0 \text {. } \] clip2net.com

8.1.3.14 Higher order differential equations

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Problem: Find the general solution of the linear homogeneous differential equation with constant coefficients: \[ y^{\prime \prime \prime}-8 y=0 \text {. } \] clip2net.com

8.1.3.15 Higher order differential equations

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Problem: Find the general solution of the linear homogeneous differential equation with constant coefficients. \[ y^{I V}+2 y^{\prime \prime}-2 y=0 \text {. } \] clip2net.com

8.1.3.16 Higher order differential equations

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Problem: Find the general solution of the linear homogeneous differential equation with constant coefficients. \[ 25 y^{I V}+10 y^{2}+y=0 \text {. } \] clip2net.com

8.1.3.17 Higher order differential equations

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Problem: Find the general solution of the third order linear inhomogeneous differential equation: \[ y^{\prime \prime \prime} \tan 5 x=5 y^{\prime \prime} \text {. } \]

8.1.3.25 Higher order differential equations

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Problem: Solve the equation using the formula that reduces multiple integration to single one: \[ x y^{I V}+y^{\prime \prime \prime}=e^{x} \text {. } \]

8.1.3.51 Higher order differential equations

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Problem: Find the general solution of the higher order equation: \[ y^{\prime \prime \prime}-3 y^{\prime \prime}-y^{\prime}+3 y=(4-8 x) e^{x} . \]

8.1.3.55 Higher order differential equations

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Problem: Find the general solution of the higher order differential equation: \[ y^{\prime \prime \prime}-81 y^{\prime}=162 e^{9 x}+81 \sin 9 x \text {. } \]

8.1.3.61 Higher order differential equations

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Problem: Find the general solution of the higher order differential equation: \[ y^{(4)}=\cos 5 x \text {. } \]

8.1.3.66 Higher order differential equations

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Problem: Find the general solution of the corresponding homogeneous differential equation and the form of the particular solution for the higher order differential equation: \[ y^{(4)}-2 y^{\prime \prime \prime}+y^{\prime}-2 y=\left(x^{2}+x\right) e^{2 x}+1 \text {. } \] clip2net.com

8.1.3.68 Higher order differential equations

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Problem: Find the general solution of the differential equation: \[ x^{2} y^{\prime \prime \prime}-2 y^{\prime}=0 \text {. } \]

8.1.3.71 Higher order differential equations

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Problem: Find the general solution of the given equations, knowing their particular solutions. In those problems where the particular solution is not given, we can look for it by selecting, for example, in the form of an exponential function \( y_{1}=e^{a x} \) or algebraic polynomial \( y_{1}=x^{n}+a x^{n-1}+b x^{n-2}+\cdots \). \[ \begin{array}{l} \left(x^{2}-2 x+3\right) y^{\prime \prime \prime}-\left(x^{2}+1\right) y^{\prime \prime}+2 x y^{\prime}-2 y=0, \\ y_{1}=x, y_{2}=e^{x} . \end{array} \]

8.1.3.95 Higher order differential equations

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