Question

Find the general solution of the following equation. Express the solution explicitly as a function of the independent variat dx =y (8x? +1) y=

Answer 1

In this question, we substitute y = a.e^(f(x)). Applying the above substitution will make the equation easier to solve as some complicated terms cancel from both sides of the equality sign.

a constant and fla SOLUTION is - 9 a efloo y ( 8 002 + 1) da [ d Jo Therefore, y = aeflix) is some function of me. We can substitute y with such a function because functions of the form over RR are bjective there DCEIR ). Now, it is given that dy -) d (aefroes) = a eflac) (8x2+1) (aef(x)] Joe =) a eflse . (f(x)) = a ef(*)(8522+1) (using chain ouke. Since the above equation is true for every real number ve, we can cancel aefcae from both sides. => dffo)) - 802+1 doc So, f(x) 5(83244 (²41) doc [Defining integral =faldx + de antiderivative = (3 0 r k ) + (x + 2) [ki, kz are constants. = 8 x3 + x + c [ c = kt kz is a constant] (5x+x+) y = [Where A = a. e da 3 a. е. Thus, 023+ A. el oc)

In the above solution for y, 'A' can be any real number.