3Solving second order differential equations1. Solve the following homogenous differential equations using two methods: 1) x(t) = et substitutionand 2) Matrix eigenvalue method, 3) Using isoclines(a) -3x-4x=0; x(0) = 1(b) -5x+4x=0; x(0) 0=(0) = 0(0) = 12. Solve the following homogenous differential equation -4x+5x=0 where x(0) = 1, (0) = 0.3. Solve the following differential equation: +3 + 2x = t where x(0) = 1, (0) = 0. (For the particularsolution, use x(t) = At + B to substitute into the differential equation).

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Answered: 3. Solve the following differential…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 64EQ

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Write the system of differential equations given in the 1st photo in the normal form defined in the 2nd photo and solve the resulting system by means of the eigenvalues and eigen-vectors of the square matrix A.
1. (a) What are A, X and x' in matrix form x' = AX of the system of differential equations { FO (b) Find all eigenvalues and eigenvectors of A. (c) Find exponential solutions of the system of differential equations. (d) Verify your solutions. x₁ = 3x1 x2 = 5x1 + 2? x2
λ1 = -2, λ2=5 V1 = (1, -2) V2 = (4, -1) Using the above eigenvalues and/or eigenvectors, find the general solution of the following system of linear differential equations, writing the solution as an equation for x and an equation for y: x = 6x+4yy = − 2 x − 3 y
y" - 3y + 2y = f(x). (2) (a) Suppose that in Eq. (2), f(x) = 0. (i) Write down the order and degree of the equation, then find two values of a so that year is a solution of the equation. (ii) Write the differential equation as a system of first-order differential equations, then use the eigenvalue method to find the general solution of the system. (b) Suppose that in Eq.(2), f(x) = 4x. Use the method of undetermined coefficients to find a general solution for the equation. How many conditions are needed to find a particular solution for this equation?
x1 = x2 – 4x + 4x3 %3D 4 x½ = -3x2 - V3x, + V3x1 +X3 x3 = 7x2 - 2x1 Find the eigenvalues and eigenvectors by using the above equations.

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3. Solve the following differential equation: +32 + 2x = t where x(0) = 1, (0) = 0. (For the particular solution, use x(t) = At + B to substitute into the differential equation).