Question

\(\left(\mathrm{y}-\frac{\mathrm{xdy}}{\mathrm{d} x}\right)=3\left(1-\mathrm{x}^{2} \frac{\mathrm{dy}}{\mathrm{dx}}\right)\)

Short answer

Based on the given solution steps, identify the type of the given first-order differential equation and the solution y(x). The given first-order differential equation can be identified as a first-order linear inhomogeneous equation of the form: \(\frac{\mathrm{dy}}{\mathrm{dx}} + P(x)y = Q(x)\) The solution of this differential equation is given by: \(y(x) = C\sqrt[3]{3x^2} - \frac{1}{\sqrt[3]{9}}\) where C is the constant of integration.

Step-by-step solution

Rewrite the equation in standard form

To make the equation easier to work with, rewrite it in the standard form for first-order differential equations. We will do this by moving terms involving y and its derivative to one side of the equation. \(\frac{\mathrm{xdy}}{\mathrm{dx}} + \mathrm{x}^{2} \frac{\mathrm{dy}}{\mathrm{dx}}\mathrm{3} = 3\mathrm{y} - 3\)

Combine similar terms

Now, combine terms involving the derivative \(\frac{\mathrm{dy}}{\mathrm{dx}}\) to simplify the equation. \(\frac{\mathrm{dy}}{\mathrm{dx}}(\mathrm{x} + 3\mathrm{x}^{2}) = 3\mathrm{y} - 3\)

Rearrange the equation

Rearrange the equation, solving for \(\frac{\mathrm{dy}}{\mathrm{dx}}\). This will give us a clearer view of the differential equation we need to solve. \(\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3\mathrm{y} - 3}{\mathrm{x} + 3\mathrm{x}^{2}}\)

Identify the type of differential equation

At this point, we can identify the differential equation as a first-order linear inhomogeneous equation of the form: \(\frac{\mathrm{dy}}{\mathrm{dx}} + P(x)y = Q(x)\) Our specific equation has the following functions for P(x) and Q(x): \(P(x) = \frac{-3}{\mathrm{x} + 3\mathrm{x}^{2}}\) \(Q(x) = \frac{3}{\mathrm{x} + 3\mathrm{x}^{2}}\)

Find the integrating factor

We will now find an integrating factor, which is a function μ(x) that satisfies the following condition: \(\frac{\mathrm{d\mu}}{\mathrm{dx}} = P(x)\mu(x)\) For our given equation, we have: \(\frac{\mathrm{d\mu}}{\mathrm{dx}} = \frac{-3}{\mathrm{x} + 3\mathrm{x}^{2}}\mu(x)\) Now, integrate both sides of the integrating factor equation with respect to x: \(\mu(x) = e^{\int \frac{-3}{\mathrm{x} + 3\mathrm{x}^{2}} \mathrm{dx}}\)

Solve for the integrating factor

To solve for the integrating factor, we need to integrate the given function. After integrating, we get: \(\mu(x) = \frac{1}{\sqrt[3]{3x^2}}\)

Multiply the differential equation by the integrating factor

Now that we have the integrating factor, we multiply it with both sides of the given differential equation: \(\frac{1}{\sqrt[3]{3x^2}}\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{1}{\sqrt[3]{3x^2}}\cdot\frac{-3}{\mathrm{x} + 3\mathrm{x}^{2}}y = \frac{1}{\sqrt[3]{3x^2}}\cdot\frac{3}{\mathrm{x} + 3\mathrm{x}^{2}}\)

Solve the equation

Finally, solve the equation found in step 7: \(\int\left(\frac{1}{\sqrt[3]{3x^2}}\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{1}{\sqrt[3]{3x^2}}\cdot\frac{-3}{\mathrm{x} + 3\mathrm{x}^{2}}y\right)\mathrm{dx} = \int\frac{1}{\sqrt[3]{3x^2}}\cdot\frac{3}{\mathrm{x} + 3\mathrm{x}^{2}}\mathrm{dx}\) After integrating, we obtain the solution for y(x): \(y(x) = C\sqrt[3]{3x^2} - \frac{1}{\sqrt[3]{9}}\) Here, C is the constant of integration, which can be determined using an initial condition if provided.