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Problem 427

Find the general solution to each of the following differential equations: \([1] \mathrm{y}^{\prime}-\mathrm{y}=0 .\) [2] \(\mathrm{y}^{\prime \prime}+4 \mathrm{y}^{\prime}-\mathrm{y}=0\). [3] \(\mathrm{y}^{\prime \prime}+8 \mathrm{y}^{\prime}+16 \mathrm{y}=0\)

Expert verified

The general solutions for the given differential equations are:
1. \(y = Ce^x\)
2. \(y(x) = A e^{(-2 + \sqrt{5})x} + B e^{(-2 - \sqrt{5})x}\)
3. \(y(x) = (A + Bx)e^{-4x}\)

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Chapter 18

Find the general solution of the following system of linear differential equations: $$ \begin{aligned} &\mathrm{x}_{1}^{\prime}=4 \mathrm{x}_{1}+\mathrm{x}_{2} \\ &\mathrm{x}_{2}^{\prime}=3 \mathrm{x}_{1}+2 \mathrm{x}_{2} \end{aligned} $$

Chapter 18

(a) Write the system $$ \begin{aligned} &\mathrm{x}_{1}^{\prime}=3 \mathrm{x}_{1} \\ &\mathrm{x}_{2}^{\prime}=-2 \mathrm{x}_{2} \\ &\mathrm{x}_{3}^{\prime}=4 \mathrm{x}_{3} \end{aligned} $$ in matrix notation. (b) Solve the system. (c) Find a solution of the system which satisfies the initial conditions \(\mathrm{x}_{1}(0)=3, \mathrm{x}_{2}(0)=5\), and \(\mathrm{x}_{3}(0)=7\)

Chapter 18

Find the solution of, the following system $$ \begin{aligned} &\dot{\mathrm{x}}_{1}=\mathrm{x}_{2} \\ &\dot{\mathrm{x}}_{2}=-2 \mathrm{x}_{1}-3 \mathrm{X}_{2} \end{aligned} $$

Chapter 18

Solve the following initial value problem: $$ \begin{aligned} &\mathrm{x}_{1}^{\prime}=2 \mathrm{x}_{1}+\mathrm{x}_{2}+3 \mathrm{x}_{3} \\ &\mathrm{x}_{2}^{\prime}=2 \mathrm{x}_{2}-\mathrm{x}_{3} \\ &\mathrm{x}_{3}^{\prime}=2 \mathrm{x}_{3} \end{aligned} $$ Initial conditions: $$ \mathrm{x}_{1}(0)=1 $$ $$ \begin{aligned} &\mathrm{x}_{2}(0)=2 \\ &\mathrm{x}_{3}(0)=1 \end{aligned} $$

Chapter 18

Find the general solution to the system $$ \begin{aligned} &\mathrm{x}_{1}^{\prime}=\mathrm{x}_{1} \\ &\mathrm{x}_{2}^{\prime}=\mathrm{x}_{1}+2 \mathrm{x}_{2} \\ &\mathrm{x}_{3}^{\prime}=\mathrm{x}_{1}-\mathrm{x}_{3} \end{aligned} $$

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