Thread: Differential equation- wordy question1- urgent

1. Differential equation- wordy question1- urgent

6. The number of people, x, in a queue at a travel centre t minutes after it opens is modelled by a differential equation dx/dt= 1.4t-0.5x for values of t up to 10. Sovlve the differential equation given that x=8 when t=o.

b. An alternative model gives the differential equation dx/dt= 1.4t-o.5x for the same values of t. Verify that x=13.6e^-0.5t+2.8t-5.6 satisfies this differential equation. Verify also that when t=0 this function takes the value 8.

2. Originally Posted by kingkaisai2
6. The number of people, x, in a queue at a travel centre t minutes after it opens is modelled by a differential equation dx/dt= 1.4t-0.5x for values of t up to 10. Sovlve the differential equation given that x=8 when t=o.
You have:

$\displaystyle \frac{dx}{dt}=1.4t-0.5x$

with initial condition $\displaystyle x=8$ when $\displaystyle t=0$.

Now rewrite the DE as:

$\displaystyle \frac{dx}{dt}+0.5x=1.4t\ \ \ \ \ \dots (1)$

and we see that we have an inhomogeneous linear ODE with constant
coefficients, and so a general solution is the sum of the general solution
of the homogeneous equation:

$\displaystyle \frac{dx}{dt}+0.5x=0\ \ \ \ \ \dots (2)$,

and any particular integral of the original equation (1).

Now the homogeneous equation (2) can be solved by a number of methods,
we can use a trial solution $\displaystyle x(t)=A e^{\alpha t}$, substitute this into
the equation and find the value of $\displaystyle \alpha$ that gives the correct answer.
Another method is to note that (2) is of variable separable type.

I will use the first of these methods, I will suppose that:$\displaystyle x(t)=A e^{\alpha t}$, then:

$\displaystyle \frac{dx}{dt}+0.5x=\alpha A e^{\alpha t} + 0.5\ A e^{\alpha t}=0$,

so if this is a solution we must have $\displaystyle \alpha=-0.5$, and the general
solution of (2) is: $\displaystyle x(t)=A e^{-0.5 t}$.

Now we need to find a particular integral for (1). Here we note that the RHS is
a multiple of $\displaystyle t$, so if $\displaystyle x=m t +c$ the LHS can be made to equal
the RHS with suitable choice of $\displaystyle m$ and $\displaystyle c.$, so substituting this into (1) we get:

$\displaystyle \frac{dx}{dt}+0.5x=m+0.5(mt+c)=1.4 t$.

So equating the coefficient of $\displaystyle t$ in the above gives $\displaystyle 0.5 m=1.4$, so $\displaystyle m=2.8$,
and the constant terms gives $\displaystyle m+0.5 c =0$, or $\displaystyle c=-5.6$.

Hence a particular integral of (1) is:

$\displaystyle x(t)=2.8 t-5.6$.

Combining this with the general solution of (2) found earlier we get the
general solution of (1) is:

$\displaystyle x(t)=A e^{-0.5 t}+2.8 t-5.6\ \ \ \ \ \dots (3)$.

Now we need to fit the initial conditions. Putting $\displaystyle t=0$ into (3)
we have:

$\displaystyle x(0)=A -5.6=8$

which we solve to find that $\displaystyle A=13.6$ is applicable for this problem.

RonL

3. Originally Posted by kingkaisai2
An alternative model gives the differential equation dx/dt= 1.4t-o.5x for the same values of t. Verify that x=13.6e^-0.5t+2.8t-5.6 satisfies this differential equation. Verify also that when t=0 this function takes the value 8.
Do the last part first, substitute $\displaystyle t=0$ into the equation:

$\displaystyle x(t)=13.6 e^{-0.5 t}+2.8 t-5.6\ \ \ \ \ \dots(A)$

to get:

$\displaystyle x(0)=13.6 e^{-0.5\times 0}+2.8\times 0-5.6=13.6-5.6=8$,

as required.

Now to verify that (A) satisfies the DE we differentiate $\displaystyle x(t)$, and check
that this derivative equals $\displaystyle 1.4 t-0.5x$, which I leave as an exercise for the reader.

RonL