MTH418 Assessment Item 1

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y1′ = 7y1 − 2y2
y2′ = 4y1 + 3y2

SKU: Math009009

Question 1
Part 1
Write the following second order homogeneous differential equation as a system of first-order
differential equations.
−3y” − 5y’ + 2y = 0

Part 2
Find the general solution to your system of first order differential equations determined in
part 1 above.

Part 3
Hence write down the general solution to the original differential equation in part 1.
.

Question 2
For both of the following systems of differential equations:
 Find the real general solution.
 Determine the critical point.
 Determine the type of the critical point.
 Determine the stability of the critical point.

Part 1  
y1′ = 7y1 − 2y2
y2′ = 4y1 + 3y2

Part 2
y1′ = 8y1 − ?y2
y2′ = y1 + 10y2

Question 3 
Find the location, type and stability of all critical points by linearization of the following
nonlinear homogeneous system of differential equations.
y1′ = y2 − y22
y2′ = y1 − y12

Question 4 
Find the general solution for the following linear non-homogeneous system of differential
equations.
y1′ = 4y1 − 8y2+ 2 cosh(t)
y2′ = 2y1− 6y2 + cosh(t) + 2 sinh(t)

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