Find the general solution of the differential equation dy/dx-4y= -5y^2
isn't this a Bernoulli equation?
http://www.math.ksu.edu/math240/math240.f09/chap1part2.pdf
To find the general solution of the given differential equation, dy/dx - 4y = -5y^2, we can utilize the method of separable variables. Here are the steps:
Step 1: Rearrange the equation to separate the variables. In this case, we need to move all the y terms to one side and all the x terms to the other side:
dy/dx = (-5y^2) + 4y
Step 2: Factor out the y terms on the right-hand side:
dy/dx = y(-5y + 4)
Step 3: Separate the variables by dividing both sides of the equation by (-5y + 4):
(1/y) dy = (-5y + 4) dx
Step 4: Integrate both sides with respect to their respective variables. Integrating both sides will eliminate the differentials:
∫ (1/y) dy = ∫ (-5y + 4) dx
The integral of (1/y) dy can be found by using the natural logarithm:
ln|y| = ∫ (-5y + 4) dx
Integrating (-5y + 4) dx with respect to x yields:
ln|y| = (-5/2)y^2 + 4x + C
Note: C is the constant of integration. It represents a family of solutions.
Step 5: Solve for y by taking the exponential of both sides to eliminate the natural logarithm:
|y| = e^((-5/2)y^2 + 4x + C)
Since the absolute value of y was introduced when taking the exponential, we need to consider two cases:
Case 1: y > 0
y = e^((-5/2)y^2 + 4x + C)
Case 2: y < 0
-y = e^((-5/2)y^2 + 4x + C)
Therefore, the general solution of the given differential equation is given by:
y = e^((-5/2)y^2 + 4x + C) for y > 0
or
-y = e^((-5/2)y^2 + 4x + C) for y < 0
Note: The constant C can be determined using initial conditions or additional information provided in the problem.