dy-x%5E2ydx%3D0.jpeg "(1+x^3)dy-x^2ydx=0")
Differential Equation: (1+x^3)dy - x^2ydx = 0
In this article, we will explore the differential equation (1+x^3)dy - x^2ydx = 0. This equation is a type of first-order linear ordinary differential equation (ODE). We will learn how to solve it and find the general solution.
Form of the Equation
The given equation is a linear ODE of the form:
M(x,y)dx + N(x,y)dy = 0
where M(x,y) = -x^2y and N(x,y) = 1+x^3.
Solving the Equation
To solve this equation, we can use the method of separation of variables. First, we rewrite the equation in the form:
dy/dx = x^2y / (1+x^3)
Now, we can separate the variables by moving all the y terms to one side and the x terms to the other side:
∫(1/y)dy = ∫(x^2 / (1+x^3))dx
Integrating Both Sides
Next, we integrate both sides of the equation. The left-hand side is straightforward:
∫(1/y)dy = ln|y| + C1
The right-hand side requires a bit more work. We can use the substitution u = 1+x^3, which implies du/dx = 3x^2. Then, we get:
∫(x^2 / (1+x^3))dx = (1/3) ∫(1/u)du = (1/3) ln|u| + C2
Substituting back u = 1+x^3, we get:
(1/3) ln|1+x^3| + C2
Finding the General Solution
Now, we equate both integrals and solve for y:
ln|y| = (1/3) ln|1+x^3| + C
where C = C1 - C2 is the constant of integration. Taking the exponential of both sides, we get:
y = ±(1+x^3)^(1/3) * e^C
The general solution to the differential equation is:
y = ±(1+x^3)^(1/3) * C
where C is an arbitrary constant.
Conclusion
In this article, we have successfully solved the differential equation (1+x^3)dy - x^2ydx = 0 using the method of separation of variables. The general solution is y = ±(1+x^3)^(1/3) * C, where C is an arbitrary constant.
