Satisfy the Differential Equation - Linear Equation

In summary, the function satisfying the given differential equation is y = e^(3x)(3e^(2x)-9), with the initial condition y(0) = -6. The mistake in inputting the function into the online homework program was due to the use of 'x' instead of 't' as the variable.
  • #1
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Homework Statement



Find the function satisfying the differential equation:

dy/dx - 3y = 6e^(5x)

with y(0) = -6

Homework Equations



I believe this is Linear, so it is dy/dx + P(x)y = f(x)

The Attempt at a Solution



I chose -3y to be P and used it to obtain integrating factor e^(-3x). I multiplied it though and ended up with d/dx(e^(-3x)y) = 6e^(2x)

I integrated both sides and got e^(-3x)y = 3e^2x + C

Solving for C gets me -9 so the satisfying function is y=e^(3x)(3e^(2x)-9)

When I try to input this function into Webwork (online homework), it tells me it is incorrect and that the variable 'x' is not defined in this context.

I wanted to make sure I am correct or if somehow, x comes out of this equation.
 
Last edited:
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  • #2
I've confirmed that this is correct. The online program wanted a t variable, not x.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It describes the relationship between a function and its rate of change, and is commonly used in physics, engineering, and other fields to model real-world phenomena.

2. What is a linear differential equation?

A linear differential equation is a differential equation in which the dependent variable and its derivatives appear as a linear combination. This means that the equation can be written in the form of a linear polynomial, with no products or powers of the dependent variable or its derivatives.

3. How do you solve a linear differential equation?

To solve a linear differential equation, you need to first identify the type of equation (e.g. first-order, second-order, etc.) and then use a method such as separation of variables, integrating factors, or variation of parameters. Each method has its own steps and techniques, but the ultimate goal is to find a particular solution that satisfies the given equation.

4. What is the importance of solving differential equations?

Differential equations are essential in many fields of science and engineering, as they allow us to mathematically model and understand real-world processes and phenomena. By solving these equations, we can make predictions, analyze systems, and develop solutions to complex problems.

5. Are there real-life applications of linear differential equations?

Yes, there are many real-world applications of linear differential equations. Some examples include modeling population growth, analyzing electrical circuits, predicting the spread of diseases, and understanding the motion of objects in space. These equations can also be used to optimize processes and make predictions in various industries such as finance, economics, and ecology.

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