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Chapter 5: Problem 16

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

Short Answer

Expert verified
y = C_1 e^{x^5}

Step by step solution

01

Separate the Variables

Rewrite the given differential equation to separate the variables. Move all terms involving y to one side and all terms involving x to the other side: \[\frac{1}{y} \frac{d y}{d x} = 5 x^{4} \]
02

Integrate Both Sides

Integrate both sides with respect to their variables. On the left side, integrate with respect to y, and on the right side, integrate with respect to x: \[ \int \frac{1}{y} d y = \int 5 x^{4} d x \]
03

Evaluate the Integrals

Find the integrals of both sides. The integral of \(\frac{1}{y}\) with respect to y is \(\ln|y|\), and the integral of \(5 x^4\) with respect to x is \(x^5\): \[ \ln|y| = x^{5} + C \] where C is the constant of integration.
04

Solve for y

Exponentiate both sides to solve for y: \[ y = e^{x^5 + C} \] Since \(e^{C}\) is a constant, let \(e^{C} = C_1\): \[ y = C_1 e^{x^5} \]

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Key Concepts

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

Separation of Variables
In many differential equations, you will need to separate the variables to solve the problem. The given differential equation is \(\frac{dy}{dx} = 5x^4 y\). The first step is to separate the x and y terms. This is done by rearranging the equation: \(\frac{1}{y} \frac{dy}{dx} = 5 x^4\). Now, multiply both sides by dx and then by y to get: \(\frac{1}{y} dy = 5 x^4 dx\). Now, it is in a form where each side involves only one variable, ready for integration.
Therefore, the separation of variables helps to isolate each variable on different sides of the equation, making it easier to solve.
Integration
Integration is the process of finding the integral of a function, which is the reverse process of differentiation. With the separated variables we got: \(\frac{1}{y} dy = 5 x^4 dx\). Now, we integrate both sides with respect to their respective variables:
\(\begin{align*} \text{Left side:} & \ \ \ \ \text{Right side:} \ \end{align*}\). The integral of \(\frac{1}{y} dy\) is \(\text{ln}|y|\) and the integral of \(\text{5} x^4 dx\) is \(\frac{5 x^5}{5} = x^5\).
So, we have: \(\text{ln}|y| = x^5 + C\), where C is the constant of integration. Integration is a crucial step to combine the results and gather all terms into a single equation.
Exponentiation
Exponentiation is used to solve for y in the equation obtained after integration. We have: \(\text{ln}|y| = x^5 + C\). To remove the logarithm, we exponentiate both sides: \(\text{e}^{(\text{ln}|y|)} = \text{e}^{(x^5 + C)}\). This simplifies to: \(|y| = \text{e}^{x^5 + C}\). We can rewrite \(\text{e}^{C}\) as another constant, let's call it \(\text{C}_1\). Therefore, we get \(|y| = C_1 \text{e}^{x^5}\).
Since \(|y|\) denotes the absolute value of y, we can write the final solution as: \(\text{y} = \text{C}_1 \text{e}^{x^5}\).
Exponentiation helps to revert the function back to its original variable by cancelling out the logarithm.
  • Separation of Variables: Isolate different variables on either side of the equation.
  • Integration: Integrate both sides with respect to corresponding variables.
  • Exponentiation: Undo the logarithm to find the final solution.

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