Question

Solve the following differential equation (separable method):

y' = 1 / (2x + y)
A) y = x^2 + x + C
B) y = x^2 - x + C
C) y = -x^2 + x + C
D) y = -x^2 - x + C

Answer 1

To solve the separable differential equation y' = 1 / (2x + y), the variables are separated and integrated, yielding the solution y = x^2 - x + C.

Step-by-step explanation:

To solve the differential equation y' = 1 / (2x + y) using the separable method, we first separate the variables y and x:

1. Rewrite the equation as dy/dx = 1 / (2x + y).
2. Multiply both sides by (2x + y) to get (2x + y) dy = dx.
3. Separate variables by moving all terms with y to one side and those with x to the other to get y dy = 1/2 dx - x dy.
4. Now integrate both sides: ∫ y dy = 1/2 ∫ dx - ∫ x dy to get y^2/2 = x/2 - xy + C.
5. Rewrite the equation in terms of y to get y = x^2 - x + C.

The correct answer is B) y = x^2 - x + C.