Solving Linear Differential Equations: A Beginner's Guide

• ImAnEngineer
In summary, the conversation is about solving linear differential equations without using standard solutions. The individual is studying differential equations and is confused about the notation used in the book. They are specifically questioning why the equation is written as (1) instead of (2). The explanation given is that using 0.5 for the right-hand side makes the coefficient in front of p equal to 1, making the integration easier. The individual also realizes that they forgot to include the 'dt' on the right side of the equation. After a discussion, they understand why the 2 comes in during integration.

ImAnEngineer

Hey guys.

I've recently started studying differential equations. There is one thing I don't understand and of which I simply can't find an explanation.

I'm trying to solve some linear differential equations without using standard solutions.
Say we have the equation:
$$\frac{dp}{dt}=0.5p - 450$$

The next step is (according to my book):

$$(1) \frac{dp}{p-900}=\frac{1}{2} dt$$

All of the next steps that lead to the solution are clear to me. They use the chain rule to integrate, exponentiate, and get: $p=900+ce^\frac{t}{2}$.
But what I don't understand, is why they first write it in the form of eq.(1), and not as, say:
$$(2) \frac{dp}{.5p-450}=1 dt$$ ?

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It's equivalent, it's just that using .5 for the right hand side means the coefficient infront of p is simply 1 and the integration is slightly easier. If you used (2), a 2 would simply come out of the left hand side integration as well which makes up for the 1/2 on the right side.

Also, you do realize there is the 'dt' on the right side correct? It can't just disappear :).

Pengwuino said:
It's equivalent, it's just that using .5 for the right hand side means the coefficient infront of p is simply 1 and the integration is slightly easier. If you used (2), a 2 would simply come out of the left hand side integration as well which makes up for the 1/2 on the right side.

Also, you do realize there is the 'dt' on the right side correct? It can't just disappear :).
I thought it should be equivalent, but when I integrate it I get a different solution. I'm trying to integrate it again now to see how I should get the 2 on the left hand side. Thanks for your answer.

PS: I realized that I forgot the dt's when I had turned of the computer and went to bed :D

Ah, now I see why the 2 comes in. If you integrate it without the 2, then you forget to compensate for the chain rule factor (or whatever you call it). I don't know if that sentence makes sense, but I get it now :) .

1. What is a linear differential equation?

A linear differential equation is an equation that involves a dependent variable and its derivatives with respect to one or more independent variables. It can be written in the form of a polynomial, where the highest power of the dependent variable is 1.

2. Why is it important to solve linear differential equations?

Linear differential equations are used to model many physical, biological, and economic phenomena. By solving these equations, we can understand the behavior of these systems and make predictions about their future states. They are also essential in many engineering and scientific applications.

3. What are the steps to solve a linear differential equation?

The general steps to solve a linear differential equation are:

1. Identify the dependent and independent variables.
2. Differentiate the equation to eliminate any constant terms.
3. Group the terms with the dependent variable and its derivatives on one side and the remaining terms on the other.
4. Integrate both sides of the equation with respect to the independent variable.
5. Solve for the constant of integration, if necessary.

4. What are the different methods to solve linear differential equations?

The most common methods to solve linear differential equations are:

• Solving by separation of variables
• Solving by integrating factors
• Solving by using the characteristic equation
• Solving by using Laplace transforms
• Solving by using power series

5. Is it necessary to have a good understanding of calculus to solve linear differential equations?

Yes, a good understanding of calculus is necessary to solve linear differential equations. This includes knowledge of differentiation and integration techniques, as well as understanding of concepts such as slope fields and initial value problems. Without a solid foundation in calculus, it can be challenging to solve these equations accurately and efficiently.