Find the solution of the differential equation dy/dt = ky, k a constant, that satisfies the given conditions. y(0) = 50, y(5) = 100

Solution without isolating :

Solution with isolating :

Step-by-step explanation:

We will separate the variables so we can integrate both sides.

Multiply on both sides:

Divide both sides by :

Now we may integrate both sides:

The first condition says .

Using this into our equation gives us:

So now our equation is:

The second condition says .

Using this into our equation gives us:

Let's find .

Subtract on both sides:

I'm going to rewrite the left hand side using quotient rule for logarithms:

Reducing fraction:

Divide both sides by 5:

So the solution to the differential equation satisfying the give conditions is:

Most likely they will prefer the equation where is isolated.

Let's write our equation in equivalent logarithm form:

We could rewrite this a bit more.

By power rule for logarithms:

By product rule for logarithms:

Since the natual logarithm and given exponential function are inverses:

By commutative property of multiplication:

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step-by-step explanation: