Differential Equations #9: Solving x'(t) = f(y(t)), y'(t) = g(x(t))

Solving
x'(t) = f(y(t))
y'(t) = g(x(t))

Sometimes using algebra can be more efficient than using Laplace transforms in solving systems of differential equations.

Procedure - two paths we can take:

1. Take the derivative of x', x" = f'(y)
2. Solve for y' and equate it to g(x).
3. Solve for x(t).
4. Solve for y(t).

OR

1. Take the derivative of y', y" = g'(x)
2. Solve for x' and equate it to f(y).
3. Solve for y(t).
4. Solve for x(t).

Let's take some examples and see how this works. In all examples presented, x and y are functions of t.

1.
x'= 2y
y'= -2x

x" = 2y'

Solving for y':
y' = x"/2

Equate to g(x):
x"/2 = -2x
0 = x"/2 - 2x
0 = x" - 4x

We have a homogenous second order differential equation.

Characteristic Equation: λ^2 + 4 = 0
Roots: λ = ± 2i

Hence
x = C1 cos(2t) + C2 sin(2t)

Note x' = 2y.
y = x'/2
y = [d/dt C1 cos(2t) + C2 sin(2t) ] /2
y = -C1 sin(2t) + C2 cos(2t)

Solution:
x = C1 cos(2t) + C2 sin(2t)
y = -C1 sin(2t) + C2 cos(2t)


The practical approach to this type of problem is to end up working with a homogenous second order differential equation, if possible.

2.
x' = y + 1
y' = 2x


Note
x" = y'

Then
y' = x" = 2x
0 = x" - 2x

We have a homogenous second order equation where the characteristic equation is
λ^2 - 2 = 0 and λ = ±√2 (-√2 and √2).

So
x = C1 e^(√2 * t) + C2 * e^(-√2 * t)

With
x' = √2 * C1 * e^(√2 * t) - √2 * C2 * e^(-√2 * t) = y + 1
y = √2 * C1 * e^(√2 * t) - √2 * C2 * e^(-√2 * t) - 1

Solution:
x = C1 e^(√2 * t) + C2 * e^(-√2 * t)
y = √2 * C1 * e^(√2 * t) - √2 * C2 * e^(-√2 * t) - 1

If we run into a non-homogenous second order differential equation, its not the end other world, refer to Part 8 ( http://edspi31415.blogspot.com/2013/09/differential-equations-8-second-order.html) on how to deal with such equations.

The next example I purposely set it up so we are going to use variation of parameters.
3.
x' = y + 3
y' = -3x

Observe that y" = -3x' and x' = -y"/3, and

y"/3 + y + 3 = 0
y" + 3y + 9 = 0
y" + 3y = -9
y = y_h + y_p

y_h:
λ^2 + 3 = 0, λ = ± i*√3
y_h =C1 cos(t * √3) + C2 sin (t * √3)

y_p:
y1 = cos(t * √3)
y2 = sin(t * √3)
f(t) = -9
W = sin(t * √3) * d/dt cos(t * √3) - cos(t * √3) * d/dt sin(t * √3) = -√3
∫ y2 * f(t)/W dt = ∫ sin(t*√3) * -9 / -√3 dt = -3 * cos(t * √3)
∫ y1 * f(t)/W dt = ∫ cos(t*√3)*-9/-√3 dt = 3 * sin(t * √3)
y_p = cos(t * √3) * -3 * cos(t * √3) - sin(t * √3) * 3 * sin(t * √3) = -3

y = C1 cos(t * √3) + C2 sin (t * √3) - 3

We only have to do the variation of parameters once.

y' = -C1 * √3 * sin(t * √3) + C2 * √3 * cos(t * √3) = -3x
x = √3/3 * C1 * sin(t * √3) - √3/3 * C2 * cos(t * √3)

Solution:
x = √3/3 * C1 * sin(t * √3) - √3/3 * C2 * cos(t * √3)
y = C1 cos(t * √3) + C2 sin (t * √3) - 3


That concludes Part 9 of our series. Until next time,

Eddie


This blog is property of Edward Shore. 2013