Find in implicit form the general solution of differential equation dy/dx=2(e^x-e^-x)/y^2(e^x+e^-x)^4 with (y>0). I know this requires a seperation of variables but I beyond that I am confused by how.Thanks.
To find the general solution of the given differential equation, we will use the method of separation of variables. Here are the steps to solve the differential equation:
Step 1: Rearrange the equation to have all y terms on one side and all x terms on the other side:
y^2(dy/dx) = 2(e^x - e^(-x))/(e^x + e^(-x))^4
Step 2: Multiply both sides by dx to separate the variables:
y^2dy = 2(e^x - e^(-x))/(e^x + e^(-x))^4 dx
Step 3: Integrate both sides with respect to their respective variables:
∫y^2 dy = ∫2(e^x - e^(-x))/(e^x + e^(-x))^4 dx
Step 4: Evaluate the integrals on both sides:
Integrating ∫y^2 dy can be done using the power rule for integration:
∫y^2 dy = (1/3) y^3 + C1, where C1 is the constant of integration.
For ∫2(e^x - e^(-x))/(e^x + e^(-x))^4 dx, we can use the substitution u = e^x + e^(-x):
du = (e^x - e^(-x)) dx
dx = du / (e^x - e^(-x))
Substituting these values into the integral, we get:
∫2(e^x - e^(-x))/(e^x + e^(-x))^4 dx = ∫2 du/u^4
= -2/u^3 + C2, where C2 is another constant of integration.
Step 5: Substitute back the expression for u:
-2/(e^x + e^(-x))^3 + C2
Step 6: Combine the results from step 4:
(1/3) y^3 = -2/(e^x + e^(-x))^3 + C2
Step 7: Solve for y by taking the cube root of both sides:
y = ((-2/(e^x + e^(-x))^3 + C2)^(1/3)
This is the general solution of the given differential equation in implicit form. It is important to note that the constants C1 and C2 can be combined into a single constant, and you can also simplify the expression further if needed.