The equation y^''+y^'-2 y=x^2 is called a differential...

The equation $y^{\prime \prime}+y^{\prime}-2 y=x^2$ is called a differential equation because it involves an unknown function $y$ and its derivatives $y^{\prime}$ and $y^{\prime \prime}$. Find constants $A, B$, and $C$ such that the function $y=A x^2+B x+C$ satisfies this equation. (Differential equations will be studied in detail in Chapter 9.)

Transcript

00:01 Okay.

00:01 And the question it states that y equals x squared plus b x plus e.

00:05 And we want to find the values of a, b and c so our first step would be to differentiate why to give us wipe crime.

00:13 We can do this by using our power rule formula, in which the exponents is brought to the front and the remaining exponents will be subtracted by one, which is why, in this case, this too was brought to the front and the power x becomes to minus one or one.

00:30 Similarly be excess looks like saying be extra part one.

00:35 The one is brought to the front and it is subtracted by one x to the power of zero is just one, which is why it's not written here, and thus it would just be plus beat.

00:45 We also know that the, um differentiating on constant gives the serial, which is why that is not included as well.

00:52 Then we move from why prime toe, why double prime? so like we did in the first step, the one has brought to the front, and the exponents itself will be subtracted by one, which is why the result will be to a and in this case, b is a constant and differentiating a constant will give us europe.

01:14 We also know from the equation that why a double crime plus white prime minus two y equals x squared.

01:21 Now that we have all of our equations, we can plug, weaken, simply put them in.

01:26 So this is what i've done right here.

01:28 I just taking these three equations and plug them as they came into this secondary equation.

01:35 Our next step would be to spread this negative two onto every, um, uh, unto every part of this bracket and thus we at this long equation.

01:51 So now our goal is to find our values for a b and c.

01:56 So one thing that should be noticed is that there's only one x squared value on both sides of the equation.

02:05 And basically, we know that, um because there's only one x squared value on both sides.

02:11 They must be equal to each other, which means that negative to a x squared equals x squared.

02:18 So now we can find the volume eight, which is what we do here.

02:21 We know...

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