Solve the following separable equation
y'=e^2x/4y^3
To solve the separable differential equation y' = e^(2x) / (4y^3), we need to separate the variables and integrate each side.
Step 1: Separate the variables
Rearrange the equation to separate the variables. In this case, we want to get all terms involving y on one side and all terms involving x on the other side.
Start with the original equation:
y' = e^(2x) / (4y^3)
Multiply both sides by (4y^3) to move y terms to the left side:
(4y^3) * y' = e^(2x)
Step 2: Integrate both sides
We will integrate both sides of the equation with respect to their respective variables.
Integrate the left side with respect to y:
∫ (4y^3) * y' dy = ∫ e^(2x) dx
On the left side, we can recognize a chain rule in reverse since the derivative of y^4 with respect to y gives us 4y^3. We can apply this rule to integrate the left side:
∫ (4y^3) * y' dy = ∫ d(y^4) = y^4 + C1
On the right side, we can integrate e^(2x) with respect to x using the substitution method. Let u = 2x, then du = 2 dx:
∫ e^(2x) dx = (1/2) ∫ e^u du = (1/2) e^u + C2
Replace u with 2x:
(1/2) e^(2x) + C2
Therefore, the integrated equation becomes:
y^4 = (1/2) e^(2x) + C
Step 3: Solve for y
To find the solution, we need to solve for y. Take the fourth root of both sides:
y = ± ∛∛((1/2) e^(2x) + C)
This is the general solution to the separable equation y' = e^(2x) / (4y^3). The constant C accounts for the arbitrary constant of integration and the ± sign indicates that there are two possible solutions.