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Chapter 31: Problem 3

Solve the given differential equations. $$\frac{d y}{d x}=\frac{x}{y^{2}}$$

Short Answer

Expert verified
The solution is \( y = \sqrt[3]{\frac{3x^2}{2} + C} \).

Step by step solution

01

Separate the Variables

The given differential equation is \( \frac{d y}{d x} = \frac{x}{y^2} \). To separate the variables, multiply both sides by \( y^2 \) and \( dx \), obtaining: \( y^2 \, dy = x \, dx \). This allows us to integrate both sides with respect to their respective variables.
02

Integrate Both Sides

Now, integrate both sides of the equation: \( \int y^2 \, dy = \int x \, dx \). The left side yields \( \frac{y^3}{3} + C_1 \) and the right side yields \( \frac{x^2}{2} + C_2 \). Thus, the equation becomes: \( \frac{y^3}{3} = \frac{x^2}{2} + C \), where \( C = C_2 - C_1 \).
03

Solve for y

To express \( y \) in terms of \( x \), multiply both sides by 3 to get \( y^3 = \frac{3x^2}{2} + 3C \). By taking the cube root of both sides, \( y = \sqrt[3]{\frac{3x^2}{2} + 3C} \). This gives the general solution of the differential equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Separation
The method of variable separation is a powerful technique for solving first order differential equations, like our given equation \(\frac{dy}{dx} = \frac{x}{y^2}\). It involves rearranging the equation so that each variable appears on opposite sides. In this case:
  • Multiply both sides by \(y^2\) and \(dx\) to bring terms involving \(y\) and \(x\) together with their respective differentials.
  • This yields: \(y^2 \, dy = x \, dx\).
By separating the variables, each side of the equation becomes an integral in terms of one variable only. This setup allows you to tackle each side individually with integration.
Integration
Integration is the next pivotal step in solving the separated equation \(y^2 \, dy = x \, dx\). We need to find the antiderivatives:
  • For the left side: \(\int y^2 \, dy\), which results in \(\frac{y^3}{3}\).
  • For the right side: \(\int x \, dx\), solving it gives \(\frac{x^2}{2}\).
It's crucial to note that constants of integration arise on both sides, typically represented as \( C_1\) and \( C_2\). These constants can simplify to a single constant, \( C = C_2 - C_1\), consolidating expression of the antiderivatives.
General Solution
After integrating, our equation becomes \(\frac{y^3}{3} = \frac{x^2}{2} + C\). To reach the general solution, solve for \(y\):
  • Multiply both sides by 3 to get rid of the fraction on the left: \(y^3 = \frac{3x^2}{2} + 3C\).
  • Finally, take the cube root of both sides: \(y = \sqrt[3]{\frac{3x^2}{2} + 3C}\).
The resulting expression reflects the general solution. It incorporates an arbitrary constant \(C\), illustrating the family of solutions depending on initial conditions or specific contexts.

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Most popular questions from this chapter

Solve the given differential equations by Laplace transforms. The function is subject to the given conditions. A pendulum moves with simple harmonic motion according to the differential equation \(D^{2} \theta+20 \theta=0,\) where \(\theta\) is the angular displacement and \(D=d / d t .\) Find \(\theta\) as a function of \(t\) if \(\theta=0\) and \(D \theta=0.40 \mathrm{rad} / \mathrm{s}\) when \(t=0\)

Solve the given problems. Explain why more than one function can be a solution of a given differential equation. Find three different functions that are solutions of \(y^{\prime}=y\).

Solve the given problems. Find the equation relating the charge and the time in an electric circuit with the following elements: \(L=0.200 \mathrm{H}, R=8.00 \Omega\) \(C=1.00 \mu \mathrm{F},\) and \(E=0 .\) In this circuit, \(q=0\) and \(i=0.500 \mathrm{A}\) when \(t=0\)

Find the inverse transforms of the given functions of \(s\) $$F(s)=\frac{1}{s^{3}+3 s^{2}+3 s+1}$$

Solve the given differential equations by Laplace transforms. The function is subject to the given conditions. $$y^{\prime \prime}+2 y^{\prime}+y=3 t e^{-7}, y(0)=4, y^{\prime}(0)=2$$
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