- #1

Saladsamurai

- 3,020

- 7

## Homework Statement

[tex]y'=(2x+y+1)^{-1}[/tex]

## Homework Equations

Find the complimentary Yc and then use annihilator to find Yp (<-- at least that is what I am trying)

## The Attempt at a Solution

[tex]y'=(2x+y+1)^{-1}[/tex]

[tex]\Rightarrow \frac{1}{y'}=2x+y+1[/tex]

[tex]\Rightarrow \frac{1}{y'}-y=2x+1[/tex]

In operator form:

[tex]\Rightarrow (\frac{1}{D}-1)y=2x+1[/tex]

Finding Yc by characteristic equation:

[tex]\Rightarrow \frac{1}{m}-1=0[/tex]

[tex]\Rightarrow m=1[/tex]

Thus, [tex]Y_c=c_1e^x[/tex]

To find Yp, multiplying both sides of [itex](\frac{1}{D}-1)y=2x+1[/itex] by the operator [itex]D^2[/itex]:

[tex]\Rightarrow D-D^2=D(1-D)=0[/tex]

[tex]\Rightarrow D=0,1[/tex]

[tex]\Rightarrow Y_p=A+Bxe^x[/tex] due to the repeated root.

Does this look okay so far?

If so, I just find Y'p and Y''p and plug them back into [tex]\Rightarrow \frac{1}{y'}=2x+y+1[/tex]

and compare coefficients of like powers of x right?

Is there a better way to do this? (keep in mind it is still early on in the class, so we may not have learned the easier way if so)