Video Transcript
Is the differential equation dπ¦ by
dπ₯ plus π₯ root π¦ equals π₯ squared linear?
A linear differential equation is
one which can be expressed as a linear polynomial of the unknown function, in this
case π¦ and its derivatives. What this means is that the only
powers of the unknown function and each derivative that appear in the equation are
one or zero if the equation does not contain that order derivative. And also each derivative and the
function itself are multiplied by functions of π₯ only.
So, for example, the equation two
times dπ¦ by dπ₯ plus four π₯π¦ equals three π₯ would be an example of a linear
differential equation. Because the power of both π¦ and
dπ¦ by dπ₯ is one, and theyβre each multiplied by a function of π₯ only. Whereas the equation two times dπ¦
by dπ₯ plus four π₯ over π¦ equals three π₯ is nonlinear as in the second term, the
power of π¦ is negative one. The equation four π₯ d two π¦ by
dπ₯ squared plus two π¦ dπ¦ by dπ₯ equals seven is also nonlinear as in the second
term, we see that dπ¦ by dπ₯ is multiplied by a function of π¦, not a pure function
of π₯.
More formally, we can say that a
differential equation is linear if it can be expressed in the form shown on the
screen. Each πth order derivative of π¦
and the function π¦ itself is multiplied by a polynomial in π₯ only. So, letβs consider the differential
equation that weβve been given. And we can see that it includes the
square root of π¦. Now, another way of expressing the
square root of π¦ is as π¦ to the power of one-half. And hence, this differential
equation is nonlinear, as the power of π¦ is not equal to one.
Now, notice that it isnβt the
presence of the π₯ squared term on the right-hand side that makes this differential
equation nonlinear. π₯ is the independent variable in
this equation. And it is only the powers of the
dependent variable and its derivatives, thatβs π¦ and dπ¦ by dπ₯ and so on, that
must all be equal to one in order to make the equation linear.
