Thread: Differential equation - general and parictular solution

1. Differential equation - general and parictular solution

Hi!

I have been given an equation as following:

$\displaystyle y^{//}-y^{/}+\frac{1}{4}y = x-4$

Which I am to find the general $\displaystyle y^g_n$ and a particular solution $\displaystyle y^p_n$

I have found that this gives a general solution:

by
$\displaystyle y^g_n = r^2-r+\frac{1}{4}$

One solution = $\displaystyle \frac{1}{2}$

Which gives:
$\displaystyle y^g_n = Ce^\frac{x}{2} + Dxe^\frac{x}{2}$

I am then to find the particular solution $\displaystyle y^p_n$

How to proceed is then my question...

By setting:

$\displaystyle y^p_n = an+b$

and calculate from there?

How would that look?

2. Hi jokke22.

Originally Posted by jokke22
I am then to find the particular solution $\displaystyle y^p_n$

How to proceed is then my question...

By setting:

$\displaystyle y^p_n = an+b$

and calculate from there?

How would that look?
$\displaystyle y\ =\ ax+b$

$\displaystyle y'\ =\ a$

$\displaystyle y''\ =\ 0$

So $\displaystyle y''-y'+\frac14y\ =\ -a+\frac{ax+b}4\ =\ x-4.$ Then just equate coefficients.

3. Originally Posted by TheAbstractionist
Hi jokke22.
$\displaystyle y\ =\ ax+b$

$\displaystyle y'\ =\ a$

$\displaystyle y''\ =\ 0$
So $\displaystyle y''-y'+\frac14y\ =\ -a+\frac{ax+b}4\ =\ x-4.$ Then just equate coefficients.

Hmmm... I think I misinterpret what I am to do...

I end up with...

$\displaystyle \frac{-4a+ax+b}{4} = x-4$

Grateful if you could show me...

4. Originally Posted by jokke22
I end up with...

$\displaystyle \frac{-4a+ax+b}{4} = x-4$
$\displaystyle \frac a4\ =\ 1$

$\displaystyle \frac{b-4a}4\ =\ -4$

5. If two polynomials are the same for all x, you can equate individual coefficients. That is what The Abstractionist did.

You can also set x to any two values and then solve the resulting two equations for a and b. For example, since $\displaystyle \frac{-4a+ax+b}{4} = x-4$, taking x= 0 gives $\displaystyle \frac{-4a+ b}{4}= -4$ or -4a+ b= -16. Taking x= 1 gives $\displaystyle \frac{-4a+ a+ b}{4}= 1- 4$ or $\displaystyle \frac{-3a+ b}{4}= -3$ so $\displaystyle -3a+ b= -12$. Those are easy to solve.

6. Okey! I think I get it know... Thank you for the help!

However, how would this then look if you were to put a solution where,

$\displaystyle y_n= y^g_n + y^p_n$

$\displaystyle = Ce^\frac{x}{2}Dxe^\frac{x}{2} +$

7. Maybe I get it wrong... Since $\displaystyle \frac{a}{4}=1$ and $\displaystyle \frac{b-4a}{4}=-4$

The particular solution is:

$\displaystyle Ce^\frac{1}{2} + Dxe^\frac{-4}{2}$