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olchik [2.2K]
3 years ago
14

Consider the system of differential equations dxdt=−4ydydt=−4x. Convert this system to a second order differential equation in y

by differentiating the second equation with respect to t and substituting for x from the first equation. Solve the equation you obtained for y as a function of t; hence find x as a function of t. If we also require x(0)=4 and y(0)=5, what are x and y?
Mathematics
1 answer:
koban [17]3 years ago
7 0

\dfrac{\mathrm dy}{\mathrm dt}=-4x\implies x=-\dfrac14\dfrac{\mathrm dy}{\mathrm dt}\implies\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac14\dfrac{\mathrm d^2y}{\mathrm dt^2}

Substituting this into the other ODE gives

-\dfrac14\dfrac{\mathrm d^2y}{\mathrm dt^2}=-4y\implies y''-16y=0

Since x(t)=-\dfrac14y'(t), it follows that x(0)=-\dfrac14y'(0)=4\implies y'(0)=-16. The ODE in y has characteristic equation

r^2-16=0

with roots r=\pm4, admitting the characteristic solution

y_c=C_1e^{4t}+C_2e^{-4t}

From the initial conditions we get

y(0)=5\implies 5=C_1+C_2

y'(0)=16\implies-16=4C_1-4C_2

\implies C_1=\dfrac12,C_2=\dfrac92

So we have

\boxed{y(t)=\dfrac12e^{4t}+\dfrac92e^{-4t}}

Take the derivative and multiply it by -1/4 to get the solution for x(t):

-\dfrac14y'(t)=\boxed{x(t)=-\dfrac12e^{4t}+\dfrac92e^{-4t}}

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