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Question #349893

dx dt=2x-4y

dx dt=2x+4y

1
2022-06-14T13:06:37-0400
"\\dfrac{dx}{dt}=2x-4y""\\dfrac{dy}{dt}=2x+4y"

"Dx-2x+4y=0""Dy-2x-4y=0"

"(D-2)x+4y=0""-2x+(D-4)y=0"

Multiply first equation by "2" and second equarion by "D-2"

"2(D-2)x+8y=0""-2(D-2)x+(D-2)(D-4)y=0"

"-2(D-2)x+(D-2)(D-4)y"

"+(-2(D-2)x+(D-2)(D-4)y)=0"

Simplify

"D^2-6D+16y=0"

Auxiliary equation

"r^2-6r+16=0"

"r=3\\pm i\\sqrt{7}"

"y(t)=c_1e^{3t}\\cos(\\sqrt{7}t)+c_2e^{3t}\\sin(\\sqrt{7}t)"

Then

"\\dfrac{dy}{dt}=3c_1e^{3t}\\cos(\\sqrt{7}t)+3c_2e^{3t}\\sin(\\sqrt{7}t)"

"-\\sqrt{7}c_1e^{3t}\\sin(\\sqrt{7}t)+\\sqrt{7}c_2e^{3t}\\cos(\\sqrt{7}t)"

Substitute

"\\dfrac{dy}{dt}=2x+4y"

"x=-2y+\\dfrac{1}{2}\\dfrac{dy}{dt}"

"x=-2c_1e^{3t}\\cos(\\sqrt{7}t)-2c_2e^{3t}\\sin(\\sqrt{7}t)"

"+\\dfrac{3}{2}c_1e^{3t}\\cos(\\sqrt{7}t)+\\dfrac{3}{2}c_2e^{3t}\\sin(\\sqrt{7}t)"

"-\\dfrac{\\sqrt{7}}{2}c_1e^{3t}\\sin(\\sqrt{7}t)+\\dfrac{\\sqrt{7}}{2}c_2e^{3t}\\cos(\\sqrt{7}t)"

"x(t)=(-\\dfrac{1}{2}c_1+\\dfrac{\\sqrt{7}}{2}c_2)e^{3t}\\cos(\\sqrt{7}t)"

"+(-\\dfrac{1}{2}c_2-\\dfrac{\\sqrt{7}}{2}c_1)e^{3t}\\sin(\\sqrt{7}t)"

"y(t)=c_1e^{3t}\\cos(\\sqrt{7}t)+c_2e^{3t}\\sin(\\sqrt{7}t)"

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