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Chapter 5: Problem 44

Solve. $$\frac{d y}{d x}=5 x^{4} y^{2}+x^{3} y^{2}$$

Short Answer

Expert verified
y = -\frac{1}{x^5 + \frac{x^4}{4} + C}

Step by step solution

01

Combine Like Terms

Rewrite the right-hand side of the equation by factoring out the common terms. $$\frac{dy}{dx} = y^2(5x^4 + x^3)$$
02

Separate Variables

Separate the variables by dividing both sides of the equation by $$y^2$$ and then multiplying both sides by $$dx$$. $$\frac{1}{y^2} dy = (5x^4 + x^3) dx$$
03

Integrate Both Sides

Integrate both sides of the equation. $$\int \frac{1}{y^2} dy = \int (5x^4 + x^3) dx$$
04

Perform Integration

Calculate the integral on both sides. $$-\frac{1}{y} = x^5 + \frac{x^4}{4} + C$$
05

Solve for y

Solve for $$y$$ to express the solution in terms of $$x$$ and the constant of integration $$C$$. $$y = -\frac{1}{x^5 + \frac{x^4}{4} + C}$$

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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 essential in solving first-order differential equations. It involves rearranging the equation so that each variable appears on a different side of the equation. This method is particularly useful since it prepares the equation for straightforward integration.
Let's use our given problem as an example:
ero \[\frac{d y}{d x}=5 x^{4} y^{2}+x^{3} y^{2}\]Rewriting the right-hand side, we factor out common terms:
\[\frac{dy}{dx} = y^2(5x^4 + x^3)\]Next, we separate variables by dividing both sides by \(y^2\) and multiplying both sides by \(dx\). This results in:
\[\frac{1}{y^2} dy = (5x^4 + x^3) dx\]In this form, integration becomes possible.
Integration
Integration is the process of finding the integral of a function, which is the reverse operation of differentiation. Upon successful variable separation, the next step is integrating both sides of the equation.
Integrate both sides:
\[\begin{align*} \text{Left side:} & \frac{1}{y^2} dy = \theta \ \text{Right side:} & (5x^4 + x^3) dx \ \text{integral:} & \theta -\frac{1}{y} = \theta + x^5 + \frac{x^4}{4} + C \end{align*}\]Constants of integration are added when we integrate, which result from the indefinite integrals.
Differential Equation Solution
After integration, the last step is to solve the differential equation explicitly for \(y\), if possible. Let's observe the solution from the previous steps.
From the equation:
\[-\frac{1}{y} = x^5 + \frac{x^4}{4} + C \]We isolate \(y\) to express it as:
\[y = -\frac{1}{x^5 + \frac{x^4}{4} + C} \]And there you have it! We've successfully solved the first-order differential equation using the method of variable separation followed by integration.
To recap:
  • Separate the variables
  • Integrate both sides
  • Solve for the dependent variable
Integrate step-by-step and always ensure every term is managed appropriately.

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Most popular questions from this chapter

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