solve the differential equation by seperation of variables 3. dy/dx=e^(2x)/4y3
4y^3 dy=e^(2x) dx
y^4=1/2 e^2x + constant
To solve the given differential equation using separation of variables, we need to write it in the form dy/dx = g(x) * h(y), where g(x) depends only on x and h(y) depends only on y.
The given differential equation is: dy/dx = e^(2x)/(4y^3)
To separate the variables, we can rewrite the equation as follows:
4y^3 dy = e^(2x) dx
Now we can bring all terms involving y to the left side and all terms involving x to the right side:
4y^3 dy - e^(2x) dx = 0
Next, we integrate both sides of the equation with respect to their respective variables. Integrating the left side with respect to y and the right side with respect to x:
∫4y^3 dy - ∫e^(2x) dx = ∫0 dx
Integrating the left side:
∫4y^3 dy = y^4 + C1, where C1 is the constant of integration.
Integrating the right side:
∫e^(2x) dx = (1/2)e^(2x) + C2, where C2 is the constant of integration.
So the equation becomes:
y^4 + C1 - (1/2)e^(2x) - C2 = 0
Combining the constants of integration:
y^4 - (1/2)e^(2x) + C = 0, where C = C1 - C2
Thus, the general solution to the given differential equation by separation of variables is:
y^4 - (1/2)e^(2x) + C = 0, where C is an arbitrary constant.