Differential Equations Problem

In summary, to determine whether the given functions are solutions to the given differential equation, you need to substitute them into the equation and check if they satisfy the initial conditions. Only the first function, y=4e^(2x) + 2e^(-3x), satisfies both the equation and the initial conditions. The second function, y=2e^(2x) + 4e^(-3x), satisfies the equation but not the initial conditions.
  • #1
Dao Tuat
16
0
Could someone please help me with this problem:

Consider the differential equation d^2y/dx^2 + dy/dx - 6y = 0 with the initial conditions y(0)= 6 and y'(0)=2. Determine whether the following functions are solutions:

a. y=4e^(2x) + 2e^(-3x)

b. y=2e^(2x) + 4e^(-3x)

If someone could please at least help me get started on this I would appreciate it very much

Thanks,
Dao Tuat
 
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  • #2
You need to substitute the two solutions given into the differential equation, and see if they come to zero. I don't think they're asking you to solve it directly.
 
  • #3
Be sure to also check that the solutions satisfy the given initial conditions.
 
  • #4
So then neither a or b are solutions, right?
 
  • #5
Only a. is a solution because b. doesn't satisfy y'(0) = 2 (y'(0) = -8). They both fit the equation, though, as can seen below.

(16e^(2x) + 18e^(-3x)) + (8e^(2x) - 6e^(2x)) - 6*(4e^(2x) + 2e^(-3x)
(8e^(2x) + 36e^(-3x)) + (4e^(2x) - 12e^(-3x)) - 6*(2e^(2x) + 4e^(-3x))
 
Last edited:

What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to model relationships between variables that are continuously changing over time or space.

What is the difference between an ordinary and partial differential equation?

An ordinary differential equation (ODE) involves a single independent variable, while a partial differential equation (PDE) involves multiple independent variables. ODEs are used to model relationships between one variable and its derivatives, while PDEs are used to model relationships between multiple variables and their derivatives.

Why are differential equations important in science?

Differential equations are used in many areas of science, including physics, engineering, and biology, to model and understand complex systems. They allow for the prediction of future behavior based on current conditions and help to uncover hidden relationships between variables.

What is the general solution of a differential equation?

The general solution of a differential equation is a formula or set of equations that includes all possible solutions to the differential equation. It may include arbitrary constants that can be determined by applying initial or boundary conditions to find a specific solution.

What are some common methods for solving differential equations?

Some common methods for solving differential equations include separation of variables, substitution, and using integrating factors. Other techniques, such as series solutions and numerical methods, may also be used depending on the type and complexity of the differential equation.

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