Question

Consider the first order separable equation y' = 2xeˣ² - y An implicit general solution can be written in the form eʸ = ______+ C Find an explicit solution of the initial value y(0) = -2 y = _______________

Answer 1

To solve the first order separable equation y' = 2xeˣ² - y, we can separate the variables and integrate both sides. The explicit solution is eʸ = e^((2/3)eˣ² - (1/2)y² + C).

Step-by-step explanation:

To solve the first order separable equation y' = 2xeˣ² - y, we can separate the variables and integrate both sides. Rearranging the equation, we have y' + y = 2xeˣ². Now, we can separate the variables by moving the y term to one side and the x term to the other side: y' + y = 2xeˣ² - y. Next, we integrate both sides with respect to y and x:

∫(1/y)dy = ∫(2xeˣ² - y)dx. This gives us ln|y| = (2/3)eˣ² - (1/2)y² + C, where C is the constant of integration.

To find the explicit solution, we take the exponential of both sides of the equation: e^(ln|y|) = e^((2/3)eˣ² - (1/2)y² + C). Simplifying, we get eʸ = e^((2/3)eˣ² - (1/2)y² + C).