SOLUTION: please any one solve this differential equation : (2ax+x^2)dy/dx=a^2+2ax

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Question 964019: please any one solve this differential equation :
(2ax+x^2)dy/dx=a^2+2ax
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
%282ax%2Bx%5E2%29expr%28dy%2Fdx%29%22%22=%22%22a%5E2%2B2ax
dy%22%22=%22%22expr%28%28a%5E2%2B2ax%29%2F%282ax%2Bx%5E2%29%29dx
int%28dy%29%22%22=%22%22int%28expr%28%28a%5E2%2B2ax%29%2F%282ax%2Bx%5E2%29%29dx%29
y%22%22=%22%22int%28expr%28%28a%28a%2B2x%29%29%2F%282ax%2Bx%5E2%29%29dx%29
y%22%22=%22%22a%2Aint%28expr%28%28a%2B2x%29%2F%282ax%2Bx%5E2%29%29dx%29
If the numerator were the derivative of the denominator, we
would be able to integrate using int%28du%2Fu%29=ln%28abs%28u%29%29%2BC
The derivative of the denominator is 2a+2x.
The numerator a+2x only needs to have "a" added to it to become that
So in the numerator we will add "a" and subtract "a"
y%22%22=%22%22a%2Aint%28expr%28%28red%28a%29%2Ba%2B2x-red%28a%29%29%2F%282ax%2Bx%5E2%29%29dx%29

y%22%22=%22%22a%2Aint%28expr%28%282a%2B2x-red%28a%29%29%2F%282ax%2Bx%5E2%29%29dx%29
Write as the difference of two fractions and integrate each:
y%22%22=%22%22
y%22%22=%22%22
y%22%22=%22%22a%2Aint%28expr%28%282a%2B2x%29%2F%282ax%2Bx%5E2%29%29dx%29%22%22-%22%22a%2Aint%28expr%28a%2F%282ax%2Bx%5E2%29%29dx%29
Use the natural log formula on the first integral, and take the
constant "a" outside the second integral:
y%22%22=%22%22a%2Aln%28abs%282ax%2Bx%5E2%29%29%22%22-%22%22a%5E2%2Aint%28expr%281%2F%282ax%2Bx%5E2%29%29dx%29
To the side, let's complete the square in the denominator by
adding and subtrracting a2:
2ax%2Bx%5E2%22%22=%22%22x%5E2%2B2ax%2Ba%5E2-a%5E2%22%22=%22%22%28x%2Ba%29%5E2-a%5E2
y%22%22=%22%22a%2Aln%28abs%282ax%2Bx%5E2%29%29%22%22-%22%22a%5E2%2Aint%28expr%281%2F%28%28x%2Ba%29%5E2-a%5E2%29%29dx%29
To do the remaining integral we use the formula int%28+du%2F%28u%5E2-a%5E2%29+%29%22%22=%22%22expr%28+1%2F%282a%29+%29ln%28abs%28+%28u-a%29%2F%28u%2Ba%29%29+%29%2BC
y%22%22=%22%22a%2Aln%28abs%282ax%2Bx%5E2%29%29%22%22-%22%22
y%22%22=%22%22a%2Aln%28abs%282ax%2Bx%5E2%29%29%22%22-%22%22expr%28+a%2F2+%29ln%28abs%28+x%2F%28x%2B2a%29%29+%29%2BC
Edwin