Question

Advanced Mathematics

We consider the following second order differential equation:

\mathrm{d}^{2} \mathrm{y} / \mathrm{dt}^{2}=5 \mathrm{dy} / \mathrm{dt}-4 \mathrm{y} .

i) Write this equation as a system of two first order differential equations of the form:

\mathrm{d} / \mathrm{dt} \vec{x}=\mathrm{A} \cdot \bar{x}

where

\overrightarrow{\mathrm{X}}:=\left\langle\mathrm{X}_{1}, \mathrm{X}_{2}\right\rangle^{\top}

and A is a 2x2 matrix. The matrix A is equal to:

\text { ii) The eigenvalues } \lambda_{1}, \lambda_{2} \text { of the matrix } A \text { in ascending order }\left(\lambda_{1} \leq \lambda_{2}\right), \text { are equal to: }

Write the corresponding eigenvectors (V1 corresponds to lambda1 and V2 corresponds to lambda2) in their simplest form, such as their first component is 1:

The general solution of this second order differential equation has the form:

The solution satisfying this second order differential equation and the initial conditions y(0)=0,dy/dt (0)=-27has:


Answer

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