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Chapter 5: Problem 2

$$\text { Find the general solution and three particular solutions.}$$ $$y^{\prime}=6 x^{5}$$

Short Answer

Expert verified
The general solution is \( y = x^6 + C \). Three particular solutions are \( y = x^6 \), \( y = x^6 + 1 \), and \( y = x^6 - 1 \).

Step by step solution

01

Identify the type of differential equation

The given differential equation is a first-order ordinary differential equation (ODE) of the form \( y' = f(x) \) where \( f(x) = 6x^5 \).
02

Integrate both sides

To find the general solution, integrate both sides of the equation \( y' = 6x^5 \) with respect to \( x \). This gives: \[ y = \int 6x^5 \, dx \]
03

Perform the integration

Integrate \( 6x^5 \) to find: \[ y = 6 \, \int x^5 \, dx = 6 \, \frac{x^6}{6} + C = x^6 + C \] where \( C \) is the constant of integration.
04

Write the general solution

Thus, the general solution to the differential equation is: \[ y = x^6 + C \]
05

Find three particular solutions

Choose three different values for \( C \) to find particular solutions. For example:
06

Particular Solution 1

Let \( C = 0 \, then y = x^6 + 0 = x^6 \)
07

Particular Solution 2

Let \( C = 1 \, then y = x^6 + 1 \)
08

Particular Solution 3

Let \( C = -1 \, then y = x^6 - 1 \)

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Key Concepts

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

Differential Equations
A differential equation involves derivatives of a function. In our case, the differential equation is:
\( y' = 6x^5 \). This is a first-order ordinary differential equation (ODE), meaning it contains the first derivative of the unknown function, y. Differential equations help us model various natural and engineering problems where we observe how quantities change over time or space.
To solve these equations, we generally aim to find the original function that satisfies the given relationship.
Integration
Integration is the process used to find the function, y, from its derivative, y'. To solve our differential equation, we need to reverse the differentiation process. This involves integrating both sides of the equation.
The given differential equation is
\( y' = 6x^5 \)
When we integrate this with respect to x, we find the function y:

\[ y = \frac { 6x^5^}{6} x^6 + C \].Integration of \(6x^5 \) results in \(x^6 \).
To integrate, we added 1 to the exponent and divided by the new exponent. Always remember to include the constant of integration, C.
Constant of Integration
The constant of integration, denoted as C, appears because when we differentiate a constant, it becomes zero. So, we cannot tell the exact constant from the derivative alone. In our solution \( y = x^6 + C \), any value of C will satisfy the original differential equation.This constant allows us to find multiple particular solutions from one general solution.
  • Example 1: If C = 0, the particular solution is \( y = x^6 \).
  • Example 2: If C = 1, the particular solution is \( y = x^6 + 1 \).
  • Example 3: If C = -1, the particular solution is \( y = x^6 - 1 \).
In conclusion, by choosing different values for C, we get different particular solutions, but they all follow the general solution pattern.

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