Problem
If y(x) is the solution of the differential equation xdy − (y² − 4y)dx = 0, where x > 0 and y(1) = 2, and the slope of the curve y = y(x) is never zero, then find the value of 10y(√2).
Solution
The equation can be rearranged to get dx/x = dy/(y² − 4y), which is a variable separable differential equation.
We can therefore proceed by directly integrating both sides.
The RHS needs to be simplified before we can integrate. It can be written as 1/4 (dy/(y – 4) – dy/y).
Now, on integrating both sides, we get ln|x| = (1/4)(ln|y – 4| – ln|y|) + c.
This simplifies to x4 = k|y – 4|/|y|, where k is some constant.
Using y(1) = 2, we’ll get 14 = k|2 – 4|/|2|, which gives k = 1.
So, the solution to the equation is x4 = |y – 4|/|y|. This gives y = 4/(1 ± x4), i.e. two possible curves.
Since y(1) has to be 2, the required curve is y(x) = 4/(1 + x4).
Finally, 10y(√2) equals 10.4/(1 + (√2)4), which simplifies to 40/5 or 8.
Comments
The differential equation was pretty easy. A little thought was needed to figure out the sign in the denominator for y(x). But the minus sign wouldn’t have given an integer value for 10y(√2), so it could’ve been eliminated later. Plain luck!
Ideally, instead of 10y(√2), the problem should’ve asked the value of 15y(√2), which gives integer answers for both signs in the denominator. This would’ve forced the student to think a little more, rather than getting lucky.