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Chapter 5: Problem 27

Solve. $$\frac{d y}{d x}=3 y$$

Short Answer

Expert verified
The solution is \[y = K e^{3x}\], where \(K\) is an arbitrary constant.

Step by step solution

01

Recognize the Differential Equation Type

Notice that \(\frac{d y}{d x} = 3 y\) is a first-order linear differential equation.
02

Separate Variables

Rewrite the equation to separate the variables. Divide both sides by \(y\) and multiply both sides by \(dx\): \(\frac{1}{y} dy = 3 dx\).
03

Integrate Both Sides

Integrate both sides of the equation: \[\int \frac{1}{y} dy = \int 3 dx\].
04

Solve the Integrals

The left side integrates to \(\text{ln}|y|\) and the right side integrates to \(3x + C\), where \(C\) is the constant of integration. So, \(\text{ln}|y| = 3x + C\).
05

Solve for \(y\)

Rewrite the equation to solve for \(y\). Exponentiate both sides to get \[y = e^{3x + C} = e^C e^{3x}\].
06

Simplify the Solution

Let \(e^C\) be a new constant, say \(K\). Therefore, the general solution is \[y = K e^{3x}\], where \(K\) is an arbitrary constant.

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Key Concepts

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

separation of variables
First, identify that \(\frac{d y}{d x} = 3 y\) is a type of differential equation that can be solved using separation of variables.
The goal is to rearrange the equation so that each variable is on a different side of the equation. This makes it easier to integrate.
To do this, divide both sides by \(y\) and multiply both sides by \(dx\).
This results in the equation \(\frac{1}{y} dy = 3 dx\). Now, the variables are separated: \(y\) on one side, and \(x\) on the other.
This technique sets up the equation for the next step鈥攊ntegration.
integration
Now that the variables are separated, we can move on to integrating both sides of the equation.
We take the integral of \(\frac{1}{y} dy\) and the integral of \(3 dx\).
The left side integrates to \(\text{ln}|y|\).
The right side integrates to \(3x + C\), where \(C\) is the constant of integration.
Putting it all together, we get \(\text{ln}|y| = 3x + C\).
Integrating both sides helps us move closer to finding the general solution to the differential equation.
general solution
To find the general solution, we need to solve the integrated form for \(y\).
Start by exponentiating both sides to eliminate the natural logarithm. So, we have \(y = e^{3x + C}\).
Notice \((e^{3x + C} = e^C e^{3x})\).
Here, let \(e^C\) be a new arbitrary constant \(K\).
Therefore, the general solution of the differential equation \(\frac{d y}{d x} = 3 y\) is \(y = K e^{3x}\).
This solution represents a family of curves, each defined by different values of \(K\).

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