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Chapter 4: Problem 8

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

Short Answer

Expert verified
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

01

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. \(\frac{\mathrm{xdy}}{\mathrm{dx}} + \mathrm{x}^{2} \frac{\mathrm{dy}}{\mathrm{dx}}\mathrm{3} = 3\mathrm{y} - 3\)
02

Combine similar terms

Now, combine terms involving the derivative \(\frac{\mathrm{dy}}{\mathrm{dx}}\) to simplify the equation. \(\frac{\mathrm{dy}}{\mathrm{dx}}(\mathrm{x} + 3\mathrm{x}^{2}) = 3\mathrm{y} - 3\)
03

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. \(\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3\mathrm{y} - 3}{\mathrm{x} + 3\mathrm{x}^{2}}\)
04

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): \(P(x) = \frac{-3}{\mathrm{x} + 3\mathrm{x}^{2}}\) \(Q(x) = \frac{3}{\mathrm{x} + 3\mathrm{x}^{2}}\)
05

Find the integrating factor

We will now find an integrating factor, which is a function 渭(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: \(\mu(x) = e^{\int \frac{-3}{\mathrm{x} + 3\mathrm{x}^{2}} \mathrm{dx}}\)
06

Solve for the integrating factor

To solve for the integrating factor, we need to integrate the given function. After integrating, we get: \(\mu(x) = \frac{1}{\sqrt[3]{3x^2}}\)
07

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: \(\frac{1}{\sqrt[3]{3x^2}}\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{1}{\sqrt[3]{3x^2}}\cdot\frac{-3}{\mathrm{x} + 3\mathrm{x}^{2}}y = \frac{1}{\sqrt[3]{3x^2}}\cdot\frac{3}{\mathrm{x} + 3\mathrm{x}^{2}}\)
08

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.

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Most popular questions from this chapter

(a) Find the general solution \(y_{\mathrm{h}}\) of the homogeneous differential equation \(\frac{d y}{d x}+2 x y=0\) (b) Show that the general solution of the nonhomogeneous equation \(\frac{d y}{d x}+2 x y=3 e^{-x^{2}}\) is equal to the solution \(y_{b}\) in part (a) plus a particular solution to the nonhomogeneous equation.

\(y\left(\frac{d y}{d x}\right)^{2}+2 x \frac{d y}{d x}-y=0 ; \quad y(0)=\sqrt{5}\)

Solve the following differential equations: (i) \(y x^{y-1} d x+x^{y} \ln x d y=0\) (ii) \(\mathrm{ye}^{-\pi / \mathrm{y}} \mathrm{dx}-\left(\mathrm{xe}^{-\mathrm{x} / \mathrm{y}}+\mathrm{y}^{3}\right) \mathrm{dy}=0\)

Solve the following differential equations: (i) \(x d x=\left(\frac{x^{2}}{y}-y^{3}\right) d y\) (ii) \(\frac{y}{x} d x+\left(y^{3}-\ln x\right) d y=0\) (iii) \(\frac{2 x d x}{y^{3}}+\frac{y^{2}-3 x^{2}}{y^{4}} d y=0\) (iv) \(y-y^{\prime} \cos x=y^{2} \cos x(1-\sin x)\)

Find the general solution of the first order nonhomogeneous linear equation \(\mathrm{y}^{\prime}+\mathrm{p}(\mathrm{x}) \mathrm{y}=\mathrm{q}(\mathrm{x})\) if two particular solutions of it, \(\mathrm{y}_{1}(\mathrm{x})\) and \(\mathrm{y}_{2}(\mathrm{x})\), are known.
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