Linear Differential Equations, no x in p(x) or q(x)?

In summary, when solving Linear D.E. in the form y' + P(x)y=Q(x), an integrating factor e^(integral p(x) ) is used. However, if there is no x in P(x) or Q(x), it is not wrong to use P(x)= -1 for problem 1 and Q(x)=4 for problem 2. This is because even though there is no x variable, it is still a function of x and will give the same output for every x value. Therefore, it can be viewed as a constant function.
  • #1
rygza
38
0
An integrating factor e^(integral p(x) ) can be used to solve Linear D.E. in form y' + P(x)y=Q(x), but what do i do when there is no x in P(x) or Q(x)?

for instance in these problems: #1)y' - y=4(e^x) #2)y'+2xy=4

Is it wrong to use P(x)= -1 for problem 1, and Q(x)=4 for problem 2?

This is what I've been using but I can't check if it's right (answer not in back of book).
 
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  • #2
1) p(x)=-1, q(x)=4e^x & 2) p(x)=2x and q(x)=4 ( Just because there's no x doesn't mean its not a function of x, or of whatever independent variable your working with; its just for every x value u'll get the same f(x), which is obv since its constant. you can view it as f(x)=4x^0, whatever x you use you'll get 4)
 
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  • #3
adriang said:
1) p(x)=-1, q(x)=4e^x & 2) p(x)=2x and q(x)=4 ( Just because there's no x doesn't mean its not a function of x, or of whatever independent variable your working with; its just for every x value u'll get the same f(x), which is obv since its constant.)

hmm i see. Thanks for the clarification
 

Related to Linear Differential Equations, no x in p(x) or q(x)?

1. What is a linear differential equation?

A linear differential equation is a type of differential equation where the dependent variable and its derivatives appear only in a linear form. In other words, the equation can be written as a linear combination of the dependent variable and its derivatives.

2. What does it mean for there to be no x in p(x) or q(x)?

When there is no x in p(x) or q(x), it means that the coefficients of the dependent variable and its derivatives are constants. In other words, there are no x terms present in the equation.

3. How do you solve a linear differential equation with no x in p(x) or q(x)?

To solve a linear differential equation with no x in p(x) or q(x), you can use the method of undetermined coefficients. This involves guessing a solution based on the form of the equation and then finding the values of the coefficients through substitution. Another method is variation of parameters, where you assume the solution has the form of a linear combination of known functions and solve for the coefficients using the equation.

4. Can a linear differential equation have more than one solution when there is no x in p(x) or q(x)?

Yes, a linear differential equation with no x in p(x) or q(x) can have multiple solutions. This is because the equation is a second-order differential equation, meaning it has two arbitrary constants. Therefore, there can be an infinite number of solutions, each with different values for the constants.

5. What are some real-world applications of linear differential equations with no x in p(x) or q(x)?

Linear differential equations with no x in p(x) or q(x) can be used to model various physical and biological systems, such as population growth, radioactive decay, and electrical circuits. They are also commonly used in engineering and economics to analyze and predict the behavior of systems.

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