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Chapter 41: Problem 1065

Find the general solution of: \((\mathrm{dy} / \mathrm{dx})=\mathrm{x}^{2} \mathrm{y}^{3}\).

Short Answer

Expert verified
The general solution of the given differential equation is \(y = \pm\sqrt{\frac{1}{-\frac{2}{3}x^3 - 2C}}\).

Step by step solution

01

Separate the Variables

First, we shall write the given equation in terms of differential elements, like this: \(dy = x^2y^3 dx \) Now, divide both sides by \(y^3\), to separate the variables: \(\frac{1}{y^3} dy = x^2 dx\)
02

Integrate Both Sides

Now, we'll integrate both sides of the equation: \(\int \frac{1}{y^3} dy = \int x^2 dx\) The integration of both sides is quite straightforward. On the left side, we raise the exponent of y by 1 and divide by the new exponent, while on the right side, we do the same for x: \(-\frac{1}{2y^2} = \frac{1}{3}x^3 + C\) Where C is the constant of integration.
03

Rearrange the Equation and Solve for y

Now, we need to rearrange the equation and get y as the subject: \(-\frac{1}{2y^2} - C = \frac{1}{3}x^3\) Multiply both sides by -2 to eliminate the negative sign: \(\frac{1}{y^2} + 2C = -\frac{2}{3}x^3\) We have the equation in the form \(\frac{1}{y^2} = -\frac{2}{3}x^3 - 2C\). Now, let's take the reciprocal of both sides: \(y^2 = \frac{1}{-\frac{2}{3}x^3 - 2C}\) To completely solve for y, take the square root of both sides: \(y = \pm\sqrt{\frac{1}{-\frac{2}{3}x^3 - 2C}}\) This is the general solution of the given differential equation.

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Most popular questions from this chapter

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