Is the differential equation d𝑦 by
d𝑥 plus 𝑥 root 𝑦 equals 𝑥 squared linear?
A linear differential equation is
one which can be expressed as a linear polynomial of the unknown function, in this
case 𝑦 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 𝑥 only.
So, for example, the equation two
times d𝑦 by d𝑥 plus four 𝑥𝑦 equals three 𝑥 would be an example of a linear
differential equation. Because the power of both 𝑦 and
d𝑦 by d𝑥 is one, and they’re each multiplied by a function of 𝑥 only. Whereas the equation two times d𝑦
by d𝑥 plus four 𝑥 over 𝑦 equals three 𝑥 is nonlinear as in the second term, the
power of 𝑦 is negative one. The equation four 𝑥 d two 𝑦 by
d𝑥 squared plus two 𝑦 d𝑦 by d𝑥 equals seven is also nonlinear as in the second
term, we see that d𝑦 by d𝑥 is multiplied by a function of 𝑦, not a pure function
More formally, we can say that a
differential equation is linear if it can be expressed in the form shown on the
screen. Each 𝑛th order derivative of 𝑦
and the function 𝑦 itself is multiplied by a polynomial in 𝑥 only. So, let’s consider the differential
equation that we’ve been given. And we can see that it includes the
square root of 𝑦. Now, another way of expressing the
square root of 𝑦 is as 𝑦 to the power of one-half. And hence, this differential
equation is nonlinear, as the power of 𝑦 is not equal to one.
Now, notice that it isn’t the
presence of the 𝑥 squared term on the right-hand side that makes this differential
equation nonlinear. 𝑥 is the independent variable in
this equation. And it is only the powers of the
dependent variable and its derivatives, that’s 𝑦 and d𝑦 by d𝑥 and so on, that
must all be equal to one in order to make the equation linear.