Ayers Differential Equations Schaums Outline Chap. 2 Problem 3

Thank You for answering me.
"Show that y=C1e^x+C2e^2x+x is a primitive of the differential equation d^2y/dx^2-3dy/dx+2y=2x-3 and find the equation of the integral curve through the points (0,0) and (1,0).
Substitute y=C1e^x+C2e^2x + x, dy/dx=C1e^x+2C2^2x + 1, d^2y/dx^2=C1e^x+4C2e^2x in the differential equation to obtain C1e^x+4C2e^2x -3(C1e^x + 2C2e^2x +1) + 2(C1e^x+C2e^2x +x) = 2x-3 .
When x=0, y=0: C1+C2 =0 . When x=1, y=0 : C1e+C2e^2=-1 .
Then C1=-C2=1/(e^2-e) and the required equation is y=(x+e^x-e^2x)/(e^2-e) . " This was direct quote from the book.
Now with the software I'm using and paper and pencil I have everything except do not understand 'C1=-C2=1/(e-e^2)' , I have learned that perhaps the answer is 'undetermined coefficient method' which I do not know . Does anybody have any ideas ? Thanks a lot for reading this. JP
 
Joseph P. said:
except do not understand 'C1=-C2=1/(e-e^2)' ,
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This seems pretty straightforward: you have a system of 2 linear equations with 2 unknowns [imath]C_1[/imath] and [imath]C_2[/imath]. It might look slightly more natural if instead of [imath]C_1[/imath] and [imath]C_2[/imath] use use, say, [imath]u[/imath] and [imath]v[/imath].
 
I looked for 2 equates in 2 vars but I didn't see it , Thanks I will check again
 
I also did solve what I saw was 2 equates in 2 vars but didn't come up with C1=-C2=1/e-e^2
 
Thanks a lot everybody, I thought you could use the primitive y=C1e^x+C2e^2x+x for x=1 y=0 would be: -1=C1e^1+C2e^2 as one equate and then the 1st order differentiated as the 2nd which was C1e^x+2C2e^2x+1 for x=1 y=0 was -1=C1e+2C2e^2 for those two using determinants I got [e,e^2;e,2e^2] = e*2e^2-e*e^2=-e^3 for the C1 numerator: [0,e^2;-1,2e^2]=0-(-e^2)=e^2 then C1=1/-e whereas for C2 [e,0;e,-1]=-e-0=-e and C2=-e/-e^3=1/e^2 evidently this is wrong using the equate from when x=0 y=0 which was C1+C2=0 against the equate for x=1 y=0 which was C1e+C2e^2=-1 thus from determinants denominator is [1,1;e,e^2] = 1*e^2-e*1=e^2-e numerator for C1[0,1;-1,e^2]=0-(-1)=0+1=1 and C1=(1/e^2-e) while numerator for C2=[1,0;e,-1]=-1-0=-1 and C2=-1/(e^2-e)
 
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