Linearity in differential equations

In summary, the equation (x2sinx + 4y) dx + x dy=0 is a linear equation with x as the dependent variable. The function or its derivative are only raised to the first power and not multiplied by each other. The answer in the book stating it is linear with y as the dependent variable may be incorrect. Additionally, for an equation to be linear, there should be no composited functions such as cos(y), ln(y), or e^y.
  • #1
Chris B
24
3

Homework Statement


Is the equation
(x2sinx + 4y) dx + x dy=0
linear
This problem also asks me to solve it, but I don't have a problem with that part.

Homework Equations


An equation is linear if the function or its derivative are only raised to the first power and not multiplied by each other.

The Attempt at a Solution


The answer in the back of the book says that it's linear with x as the dependent variable. I tried rearranging so that all x's were to the first power, but nothing doing. Is it a typo and it meant it's linear with y as the dependent variable? Because y is already by itself, and if you divide by dx then y' and y aren't multiplied by each other either so
x2sinx + 4y =-x dy/dx

Right? Have I mixed up dependent and independent somehow?
 
Physics news on Phys.org
  • #2
"The function" refers to y.
 
  • #3
vela said:
"The function" refers to y.
Fair enough, but if y is the function then x is the independent variable, right?
 
  • #4
Right.
 
  • #5
Okay, then my book's answer key is wrong. Thanks.
 
  • #6
You are welcome.
But just to add :
" An equation is linear if the function or its derivative are only raised to the first power, not multiplied by each other and not composited with other function ."

i.e. there is no cos(y), ln(y) , arctan(y'), e^y .. etc
 
  • Like
Likes Chris B

What is linearity in differential equations?

Linearity in differential equations refers to the property where the function that satisfies the equation can be expressed as a linear combination of its derivatives and independent variables. This means that the equation is proportional to its variables and can be solved using linear algebra methods.

What is the difference between linear and nonlinear differential equations?

Linear differential equations have the property of linearity, where the function can be expressed as a linear combination of its derivatives and independent variables. Nonlinear differential equations do not have this property and cannot be solved using linear algebra methods. Nonlinear equations often have complex solutions and require numerical methods for solving.

How do you determine if a differential equation is linear or nonlinear?

A differential equation is considered linear if it can be written in the form of y' + p(x)y = q(x), where p(x) and q(x) are functions of x. If the equation cannot be written in this form, it is considered nonlinear. Another way to determine linearity is to check if the equation satisfies the superposition principle, where the sum of any two solutions is also a solution.

Why is linearity important in differential equations?

Linearity is important in differential equations because it allows us to use linear algebra methods to solve them. This makes the equations easier to manipulate and analyze, and often leads to simpler, closed-form solutions. In many applications, linear differential equations are also used to model real-world systems, making the concept of linearity crucial for understanding and predicting their behavior.

What are some common applications of linear differential equations?

Linear differential equations have a wide range of applications in various fields such as physics, engineering, economics, and biology. They can be used to model population growth, heat transfer, electrical circuits, and many other systems. Additionally, many physical laws and principles are described by linear differential equations, making them an essential tool for understanding the natural world.

Similar threads

  • Calculus and Beyond Homework Help
Replies
7
Views
642
  • Calculus and Beyond Homework Help
Replies
1
Views
171
  • Calculus and Beyond Homework Help
Replies
4
Views
893
  • Calculus and Beyond Homework Help
Replies
21
Views
756
  • Calculus and Beyond Homework Help
Replies
8
Views
709
  • Calculus and Beyond Homework Help
Replies
2
Views
171
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
862
  • Calculus and Beyond Homework Help
Replies
3
Views
515
  • Calculus and Beyond Homework Help
Replies
1
Views
785
Back
Top