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Solving the Differential Equation (x+2y^3)dy/dx=y
In this article, we will solve the differential equation (x+2y^3)dy/dx=y. This is a first-order differential equation, and we will use separation of variables to find the general solution.
Given Equation
The given differential equation is:
(x+2y^3)dy/dx = y
Separation of Variables
To separate the variables, we can rewrite the equation as:
dy/dx = y / (x+2y^3)
Now, we can separate the variables by moving the dy to one side and the dx to the other side:
dy = y / (x+2y^3) dx
Integrating Both Sides
Next, we integrate both sides of the equation with respect to x:
∫dy = ∫y / (x+2y^3) dx
Simplifying the Integral
To simplify the integral, we can use the substitution u = x+2y^3. Then, du/dx = 1 and du = dx.
Substituting u into the integral, we get:
∫dy = ∫y / u du
Now, we can integrate both sides:
y = ln|u| + C
Solving for y
Substituting back u = x+2y^3, we get:
y = ln|x+2y^3| + C
General Solution
The general solution to the differential equation (x+2y^3)dy/dx=y is:
y = ln|x+2y^3| + C
where C is the constant of integration.
Conclusion
In this article, we solved the differential equation (x+2y^3)dy/dx=y using separation of variables. The general solution is y = ln|x+2y^3| + C, where C is the constant of integration.
