Variation of parameters applied to an ODE

In summary, when solving the non-homogeneous ODE via variation of parameters, it is important to divide the differential equation by the coefficient of the first derivative term, if it is not already 1. This will change the form of the integrals and ultimately give the correct particular solution.
  • #1
fluidistic
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The ODE to solve via variation of parameters is ##(1-x)y''+xy'-y=(1-x)^2##.
Knowing that ##e^x## and ##x## are solutions to the homogeneous ODE.
Now if I call ##y_1=x## and ##y_2=e^x##, the Wronskian is ##W(y_1,y_2)=e^{x}(x-1)##.
According to http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx, the particular solution of the non homogeneous ODE should be of the form ##-y_1 \int \frac{y_2 (1-x)^2}{W(y_1,y_2)} dx+y_2 \int \frac{y_1(1-x)^2}{W(y_1,y_2)}##. This gave me ##y_p(x)=-\frac{x^3}{2}-x-1## I've even checked out with wolfram alpha the evaluation of the integrals, that is the result. However the answer is ##y_p=x^2+1##.
I'm clueless on what's going on.
 
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  • #2
fluidistic said:
The ODE to solve via variation of parameters is ##(1-x)y''+xy'-y=(1-x)^2##.
Knowing that ##e^x## and ##x## are solutions to the homogeneous ODE.
Now if I call ##y_1=x## and ##y_2=e^x##, the Wronskian is ##W(y_1,y_2)=e^{x}(x-1)##.
According to http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx, the particular solution of the non homogeneous ODE should be of the form ##-y_1 \int \frac{y_2 (1-x)^2}{W(y_1,y_2)} dx+y_2 \int \frac{y_1(1-x)^2}{W(y_1,y_2)}##. This gave me ##y_p(x)=-\frac{x^3}{2}-x-1## I've even checked out with wolfram alpha the evaluation of the integrals, that is the result. However the answer is ##y_p=x^2+1##.
I'm clueless on what's going on.

If you take a careful look at that page you linked to, you'll notice that although the author writes the ODE as ##p(x)y'' + q(x)y' + r(x)y = g(x)##, he eventually states that he will assume p(x) = 1. The wikipedia page for variation of parameters similarly makes this assumption. Try diving your differential equation by (1-x) and then applying the ##-y_1 \int \frac{y_2 g(x)}{W(y_1,y_2)} dx+y_2 \int \frac{y_1g(x)}{W(y_1,y_2)}## formula, where g(x), the non-homogeneous term, is (1-x) rather than (1-x)2.
 
Last edited:
  • #3
Mute said:
If you take a careful look at that page you linked to, you'll notice that although the author writes the ODE as ##p(x)y'' + q(x)y' + r(x)y = g(x)##, he eventually states that he will assume p(x) = 1. The wikipedia page for variation of parameters similarly makes this assumption. Try diving your differential equation by (1-x) and then applying the ##-y_1 \int \frac{y_2 g(x)}{W(y_1,y_2)} dx+y_2 \int \frac{y_1g(x)}{W(y_1,y_2)}## formula, where g(x), the non-homogeneous term, is (1-x) rather than (1-x)2.

Thank you very much. I totally missed this.
 

1. What is the variation of parameters method?

The variation of parameters method is a technique used to solve a non-homogeneous linear ordinary differential equation (ODE). It involves finding a particular solution by assuming it has the same form as the homogeneous solution but with undetermined coefficients.

2. When is the variation of parameters method used?

The variation of parameters method is typically used when the non-homogeneous term in the ODE is a polynomial, exponential, or trigonometric function. It is also useful when the ODE cannot be solved using other methods such as separation of variables or integrating factors.

3. How does the variation of parameters method work?

The variation of parameters method involves finding a particular solution by assuming it has the same form as the homogeneous solution but with undetermined coefficients. These coefficients are then solved for by substituting the particular solution into the original ODE and equating coefficients of like terms.

4. What are the advantages of using the variation of parameters method?

One advantage of using the variation of parameters method is that it can be used to solve a wide range of non-homogeneous linear ODEs. It also allows for the inclusion of initial conditions, making it particularly useful for solving boundary value problems. Additionally, the method is relatively simple and straightforward to apply.

5. Are there any limitations to the variation of parameters method?

While the variation of parameters method is a powerful tool for solving ODEs, it does have some limitations. It can only be used on linear ODEs, and the non-homogeneous term must be a polynomial, exponential, or trigonometric function. Also, the method can become more complicated and time-consuming for higher-order ODEs.

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