Cauchy-Euler with x=e^t? Differential Equations (ODE)

In summary: So, with your ##y_h## and ##y_p##, I get ##y = -\frac 1 5 (x^2 + 9)(-\frac 1 5 x^2) = \frac {x^2 + 9}{5}##, which also checks.In summary, the conversation discusses using the Cauchy-Euler method with x=e^t to solve a differential equation. The individuals discuss different techniques for finding the particular solution to the nonhomogeneous problem, and there is some confusion about whether to use a substitution or the variation of parameters method. Ultimately, they come to the conclusion that both methods result in the same particular solution of -1/5 e^2t and
  • #1
kepherax
9
1
Homework Statement
Use Cauchy-Euler method with x=e^t to solve the following differential equation.
Relevant Equations
x^2y''(x)+xy'(x)-9y(x)=x^2
I'm fine with this up to a certain point, but I'm not certain if I'm using the substitution correctly. After finding the homogeneous solution do I plug in x= e^t in the original equation and then divide by e^2t to put it in standard form before applying variation of parameters so f=1, or do I just substitute so f= e^2t? The former is shown, but the latter gives a matching answer to plugging the original equation into symbolab if I were to replace their x with e^t. In videos I've watched and as per the book, you must always put the original equation into standard form and use the resulting f, but I feel like I'm missing something I can't put my finger on.

20191013_142823.jpg
 
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  • #2
kepherax said:
Homework Statement: Use Cauchy-Euler method with x=e^t to solve the following differential equation.
Homework Equations: x^2y''(x)+xy'(x)-9y(x)=x^2

I'm fine with this up to a certain point, but I'm not certain if I'm using the substitution correctly. After finding the homogeneous solution do I plug in x= e^t in the original equation and then divide by e^2t to put it in standard form before applying variation of parameters so f=1, or do I just substitute so f= e^2t? The former is shown, but the latter gives a matching answer to plugging the original equation into symbolab if I were to replace their x with e^t. In videos I've watched and as per the book, you must always put the original equation into standard form and use the resulting f, but I feel like I'm missing something I can't put my finger on.

View attachment 251084
You seem to have an error in your particular solution. If you substitute just your particular solution into the differential equation, you get ##x^2 \cdot 0 + x \cdot 0 - 9 \cdot -\frac 1 9 \ne x^2##. Since the two exponentials make up the solution to the homogeneous problem, all of the terms resulting from that part of the solution just vanish.

I got a particular solution of the nonhomogeneous problem of ##y_p = -\frac 1 5 x^2##, and this checks out.

Your solution to the homogeneous problem is fine, but for the nonhomogeneous problem, I used a different technique, assuming that the solution to ##x^2y'' + xy' - 9y = x^2## was a polynomial of the form ##y_p = c_0 + c_1x + c_2x^2##.

I'm pretty rusty on the variation of parameters technique, so I don't have any comments on what you did in the lower half of the page in the photo.
 
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  • #3
Mark44 said:
You seem to have an error in your particular solution. If you substitute just your particular solution into the differential equation, you get ##x^2 \cdot 0 + x \cdot 0 - 9 \cdot -\frac 1 9 \ne x^2##. Since the two exponentials make up the solution to the homogeneous problem, all of the terms resulting from that part of the solution just vanish.

I got a particular solution of the nonhomogeneous problem of ##y_p = -\frac 1 5 x^2##, and this checks out.

Your solution to the homogeneous problem is fine, but for the nonhomogeneous problem, I used a different technique, assuming that the solution to ##x^2y'' + xy' - 9y = x^2## was a polynomial of the form ##y_p = c_0 + c_1x + c_2x^2##.

I'm pretty rusty on the variation of parameters technique, so I don't have any comments on what you did in the lower half of the page in the photo.

Yeah, even when I use the variation of parameters method with f = e^2t, I get -1/5 e^2t as the particular solution, which works with the substitution... part of the process with Cauchy Euler is to first solve the homogeneous equation and then divide through to put in standard form for parameter variation, though, which either way (with or without sub) will result in f=1 and as pictured, so I'm confused. I may ask a tutor tomorrow morning, I saw one today but he was also rusty on ODE and couldn't clarify this for me. Thanks for trying!
 
  • #4
kepherax said:
I get -1/5 e^2t as the particular solution, which works with the substitution.
And with x = e^t, your solution is the same as mine; namely, ##y_p = -\frac 1 5 x^2##.
 

Related to Cauchy-Euler with x=e^t? Differential Equations (ODE)

1. What is Cauchy-Euler equation?

The Cauchy-Euler equation is a type of linear ordinary differential equation (ODE) that involves a power function of the independent variable and its derivatives. The general form of a Cauchy-Euler equation is ax^n y^(n) + bx^(n-1) y^(n-1) + ... + kxy' + py = 0, where a, b, ..., k are constants and n is a real number.

2. How is Cauchy-Euler equation solved?

The solution to a Cauchy-Euler equation can be found by assuming a power series solution of the form y = x^r, where r is a constant. This leads to a characteristic equation r(r-1) + br + c = 0, where b and c are constants. The roots of this equation will give the values of r, which can then be used to construct the general solution.

3. What is the significance of x=e^t in Cauchy-Euler equation?

When the independent variable x is equal to the exponential function e^t, the Cauchy-Euler equation simplifies to a constant coefficient equation. This means that the solution to the Cauchy-Euler equation with x=e^t can be found using the methods for solving constant coefficient equations, such as the method of undetermined coefficients or variation of parameters.

4. Can Cauchy-Euler equation be used to model real-world phenomena?

Yes, Cauchy-Euler equations can be used to model real-world phenomena such as population growth, radioactive decay, and heat transfer. They are commonly used in physics, engineering, and economics to describe systems that involve power functions and their derivatives.

5. Are there any limitations to the Cauchy-Euler equation?

One limitation of the Cauchy-Euler equation is that it can only be applied to linear ODEs. Nonlinear ODEs cannot be solved using the power series method, which is the key approach for solving Cauchy-Euler equations. Additionally, the solutions to Cauchy-Euler equations with repeated roots may require additional techniques, such as the method of reduction of order.

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