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.
dy/dx=2(first )/y^2 (second)
y^2 dy=2(first)dx/(second)
so lets look at the right side.
2(e^x-e^-x)/(e^x+e^-x)^4
factor out e^-x
numerator: 2e^-x(e^2x)
denominator(factor out e^-4x)
e^-4x (e^2x-1)^2
so you are left on the right side:
2*e^3x(e^2x -1)/(e^2x+1) dx
or = 2(e^5x)/(e^2x+1) dx + 2e^3x/(e^2x+2) dx
thanks Bob much appreciated. Do I then take the integral of the last line? And is the left hand side 1/3y^3 (from y^)? I do not know how this fits together. Thanks for any further help
To find the general solution of the given differential equation, let's start by rearranging the equation:
dy/dx = 2(e^x - e^-x) / y^2(e^x + e^-x)^4
Multiply both sides of the equation by y^2(e^x + e^-x)^4 to separate the variables:
y^2(e^x + e^-x)^4 dy = 2(e^x - e^-x) dx
Now, let's integrate both sides of the equation with respect to their respective variables. First, integrate the left side:
∫ y^2(e^x + e^-x)^4 dy
To integrate this expression, you can use the substitution method. Let u = e^x + e^-x.
To find du/dx, differentiate both sides of the equation with respect to x:
du/dx = d/dx (e^x + e^-x)
= e^x - e^-x
Rearrange this equation to solve for dx:
dx = du / (e^x - e^-x)
Substituting back into the integral, we have:
∫ y^2(e^x + e^-x)^4 dy = ∫ 2 du
The integration on the left side becomes:
∫ y^2 u^4 dy = 2u + C1
Now, let's integrate the right side:
∫ 2 du = 2u + C2
Therefore, the equation becomes:
∫ y^2(e^x + e^-x)^4 dy = ∫ 2 du
=> ∫ y^2 u^4 dy = 2u
Now, we have a separable equation. We can rewrite it as:
∫ y^2 dy = 2 ∫ u^4 du
Integrate both sides of the equation:
(y^3 / 3) = (u^5 / 5) + C3
Substituting back for u:
(y^3 / 3) = [(e^x + e^-x)^5 / 5] + C3
Finally, solve for y:
y^3 = [5/3 * (e^x + e^-x)^5] + C4
Taking the cube root of both sides:
y = (5/3)^(1/3) * (e^x + e^-x)^(5/3) + (C4)^(1/3)
This is the general solution in implicit form to the given differential equation.