I am having a bit of trouble finding a method of solving the differential equation of

dy/dx=f(y)+g(x)

How do we solve these sorts of equations?

Thanks,

Brad

Thanks,

I had originally wished to solve the equation of

dy/dx=y+x

I just can't seem to do this methodically (I think that a solution is y=-x-1, but there most probably are other solutions). Any help would be greatly appreciated.

Brad

Ah well that's much easier than the general case. The nice thing with your example is that the differential equation is linear in y.

You have already found one solution: y = -x - 1 (to see this works just plug in y = -x - 1 to both sides and make sure they balance).

So if you let y₁ = -x - 1 then:

dy₁ /dx = y₁ + x [equation 1]

Now suppose there exists another solution, y such that

dy/dx = y + x [equation 2]

Subtract [1] from [2]:

dy/dx - dy₁ /dx = y - y₁ + x - x

So:

d(y - y₁ )/dx = (y - y₁ )

So the function u = y - y₁ satisfies:

du/dx = u

Therefore u = Ae^x , y - y₁ = Ae^x

y = - x - 1 + Ae^x

And remember that this holds for any y that satisfies the differential equation, so the general solution is:

y = -x - 1 + Ae^x

where A is an arbitrary constant.

Hope this helps,

Michael

There is also an alternative method which can be used to solve the more general dy/dx = y.f(x) + g(x) which you might be interested in.

If dy/dx + Py = Q where P and Q are functions of x
dy/dx + Py = Q

y=e^{-òP(x) dx}ò(Q(x)e^{òP(x ' )dx '} dx

Kerwin