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Solving the ansatz equation x^2 - 4x + 3 = gives 2 equal roots where x = 3 and x = 1The general solution therefore is y = Ae^3x + Be^x where A and B are arbitrary constants
If you have studied Physics, you may be familiar with the small angle approximation cos θ ≈ 1 - θ2 / 2. This is a surprisingly accurate approximation for small values of theta (try it!), but w...
by completing the square we write the equation as (x+b/2)^2-b/2^2+c, in this case b=8 (the coefficient of x) and c=5 so we have (x+4)^2-16-5=0, which equals (x+4)^2-21=0. Now by rearranging we get (x+4)^2...
Here we can use integration by parts. Notice that ln(x) can be written as ln(x)1, so we can integrate 1 and differentiate ln(x).Then using the formula int(uv') dx = uv - int(u'v) dx, we...
It can first be observed that this differential equation is linear, so we can solve it by multiplying the whole equation by the integrating factor. As there is no coefficient in front of the dy/dx term, w...
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