Question

Suppose M and N are functions of only one variable. Is the method of solving exact equations is the best method to use? Justify your answe​

Answer 1

A first-order differential equation (of one variable) is called exact, or an exact differential, if it is the result of a simple differentiation. The equation P(x, y)y′ + Q(x, y) = 0, or in the equivalent alternate notation P(x, y)dy + Q(x, y)dx = 0, is exact if Px(x, y) = Qy(x, y).

Answer 2

Therefore, the function f( x,y) whose total differential is the left‐hand side of the given differential equation is

and the general solution is

The particular solution specified by the IVP must have y = 3 when x = 0; this condition determines the value of the constant c:

Thus, the solution of the IVP is

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