Video: Finding the General Antiderivative of a Function Using the Power Rule of Integration with Fractional Exponents

Find the solution of the differential equation (d𝑦/dπ‘₯) + 9𝑦 = 63 given that 𝑦(0) = 8.

04:20

Video Transcript

Find the solution of the differential equation d𝑦 by dπ‘₯ plus nine 𝑦 is equal to 63 given that 𝑦 nought is equal to eight.

For this type of question, the first thing we should do is to check whether we have been given a separable differential equation. This is a differential equation that can be written in the form d𝑦 by dπ‘₯ is equal to 𝑔 of π‘₯ multiplied by 𝑓 of 𝑦. An equivalent statement to this would be one over 𝑓 of 𝑦 d𝑦 by dπ‘₯ is equal to 𝑔 of π‘₯. We might also decide to define another function β„Ž of 𝑦 to make notations slightly easier, where β„Ž of 𝑦 is equal to one over 𝑓 of 𝑦.

Now, here we should be a little bit careful. Just because our equation has all of the 𝑦 terms on the left does not mean it’s already in this form. This is because we have an addition here. Consider the following side example of the differential equation d𝑦 by dπ‘₯ plus 𝑦 is equal to two π‘₯. Although it might look like this is a separable differential equation because we have a 𝑦 on the left and an π‘₯ on the right. In actual fact, no matter how much we try, we cannot express it in the correct way. This example illustrates that some equations which look like they should be separable in fact are not. Luckily, we are not dealing with one of these cases. However, our equation does require a little bit of manipulation before we proceed.

We take our equation and we subtract nine 𝑦 from both sides. We can then factorize the right-hand side since both terms have a common factor of nine. We then divide both sides by seven minus 𝑦. This gives us the following equation which does indeed match the form that we’ve shown here. Okay, we’re now gonna use a little bit of a trick. We can treat d𝑦 by dπ‘₯ somewhat like a fraction. Doing so allows us to reach a different form of equation: one over seven minus 𝑦 d𝑦 is equal to nine dπ‘₯. From this form, we can then integrate both sides of our equation. This gives us the negative of the natural logarithm of the absolute value of seven minus 𝑦 is equal to nine π‘₯ plus 𝑐, where 𝑐 is a constant. Of course, both of our integrals would have given us a constant of integration. However, we’ve chosen to combine both of these on the right-hand side into the constant, which we’ve just called 𝑐.

Okay, we’ve now found the general solution to our differential equation. However, this solution is in an implicit form. An explicit solution would be one in the form 𝑦 equals some function of π‘₯. We’ve chosen to use 𝑒 here to avoid confusion with this 𝑓 in our definition. To make things easier to manage, let’s work towards converting our general implicit solution to a general explicit solution. The first thing we can do is multiply both sides by negative one. Since we haven’t yet defined our constant, it doesn’t matter whether we have a plus 𝑐 or a minus 𝑐. So perhaps, we’ll just leave it as a plus 𝑐. Let’s clear some room to continue.

The next thing we do is take the exponential of both sides, which means raising 𝑒 to the power of both sides of our equation. Since this is the inverse of the natural algorithm on the left-hand side of the equation, we’re simply left with seven minus 𝑦. Next, we can simplify by subtracting seven from both sides and then multiplying by negative one. This gives us that 𝑦 is equal to negative 𝑒 to the power of negative nine π‘₯ plus 𝑐 add seven. At this stage, we might recognize that 𝑒 to the power of negative nine π‘₯ plus 𝑐 is equal to 𝑒 to the power of negative nine π‘₯ multiplied by 𝑒 to the power of 𝑐. Doing this gives us another way to simplify.

Together here, we have negative 𝑒 to the power of 𝑐. But since 𝑐 is currently an undefined constant, we can redefine a new constant. Let’s say capital 𝐢, which is equal to negative 𝑒 to the power of our previous lowercase 𝑐. Doing so allows us to say that 𝑦 is equal to capital 𝐢 times 𝑒 to the power of negative nine π‘₯ plus seven. We have now found a general implicit solution to our differential equation. At this stage, we’re ready to use the remaining information given by the question. The question has given us that 𝑦 of nought is equal to eight this information is our initial value and we need to find the particular solution to the differential equation for which this is true.

From this statement, we know that when π‘₯ is equal to zero, 𝑦 must be equal to eight. Another way of saying this would be that the curve which represents our particular solution will pass through the point zero, eight. Okay, we can substitute our known values into the general solution to find the value of capital 𝐢 for our particular solution. Since 𝑒 to the power of nought is equal to one, we have that eight is equal to 𝐢 times one add seven. Therefore, our constant capital 𝐢 is equal to one. Substituting this value of 𝐢 back into our general solution gives us the particular solution to our differential equation, which matches the initial value given by the question. The equation that represents our particular solution is that 𝑦 is equal to 𝑒 to the power of negative nine π‘₯ plus seven.

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