Need help for question #4 ## 4.2 ProblemsFind the general solutions of the linear systems in Problems 1 through 20. If initial conditions are given, find the particular solution that satisfies them. In Problems 1 through 6, use a computer system or graphing calculator to construct a direction field and typical solution curves for the given system.1. \( x' = -x + 3y, \, y' = 2y \)2. \( x' = x - 2y, \, y' = 2x - 3y \)3. \( x' = -3x + 2y, \, y' = -3x + 4y; \, x(0) = 0, \, y(0) = 2 \)4. \( x' = 3x - y, \, y' = 5x - 3y; \, x(0) = 1, \, y(0) = -1 \)5. \( x' = -3x - 4y, \, y' = 2x + y \)6. \( x' = x + 9y, \, y' = -2x - 5y; \, x(0) = 3, \, y(0) = 2 \)7. \( x' = 4x + \frac{1}{2}y, \, y' = -2x + \frac{1}{2}y \)### Graphs/Diagrams ExplanationFor problems 1 through 6, you are instructed to use a computer system or graphing calculator to create a direction field and identify typical solution curves for the given systems. A direction field is a graphical tool for visualizing the behavior of solutions to differential equations, showing the slope of solution trajectories at sample points in the plane. In addition to these fields, typical solution curves, which represent particular solutions based on different initial conditions, provide insight into the diverse dynamics of the system.

Name: Need help for question #4 ## 4.2 ProblemsFind the general solutions of
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Description: Need help for question #4 ## 4.2 ProblemsFind the general solutions of the linear systems in Problems 1 through 20. If initial conditions are given, find the particular solution that satisfies them. In Problems 1 through 6, use a computer system or g

The given system of the equation is: x'=3x-y, y'=5x-3y with initial conditions x(0)=1 and y(0)=-1. The matrix representation of the system is: x'y'=3-15-3xy Let A=3-15-3.Find the Eigen values A: det (Iλ-A)=λ-3-15λ+3=(λ-3)(λ+3)+5=λ2-9+5=λ2-4 det (λI - A)=0 ⇒λ2-4=0. λ2-4=0λ2=4λ=2 and λ=-2 The Eigen values are 2 and -2.Find the Eigen vectors of A: 1. Eigen vector of A for the corresponding Eigen value 2: Av=2v where v=v1v23-15-3v1v2=2v1v23v1-v25v1-3v2=2v1v23v1-v2-2v15v1-3v2-2v2=00v1-v25v1-5v2=00 Solve v1-v2=0: v1-v2=0v1=v2Choose v1=1 then v2=1 11 is the Eigen vector corresponding the to the Eigen value 2Similarly 151 is the Eigen vector corresponding to the Eigen value -2. Result: The solution of the linear system X'=AX is: c1v1eλ1t+c2v2eλ2t+⋯+cnvneλnt where λis are Eigen value and vi are the Eigen vector corresponding to it. The general solution of the given system of linear differential equation is: xy=c1e2t11+c2e-2t151xy=c1e2t11+c2e-2t151 Apply the initial conditions: x(0)y(0)=c1e2(0)11+c2e-2(0)1511-1=c111+c2151c1+15c2=1c1+c2=-1 Solve the system of equation by elimination method: c1=32 and c2=-52 The particular solution is: xy=32e2t11-52e-2t151xy=32e2t11-12e-2t15From the given problem, x'(t)=3x-y and y'(t)=5x-3y. dydx=dydtdxdt=y'x' Substitute the values in the equation: dydx=5x-3y3x-yDraw the directional field of dydx=5x-3y3x-y using graphing tool (Geogebra). The solution of the given linear differential equation with initial conditions is: xy=32e2t11-12e-2t15 The directional field of the given problem is: