5 + y= e X8)dx

1.	5 + y= e X8)dx
Answer» The integrating factor is e^(integral(1 dx)) = e^x. Multiply both sides by e^x:e^x dy/dx + e^x y = e^(2x)==> (d/dx)(e^x * y) = e^(2x), by the product rule for derivatives. Integrate both sides: e^x * y = e^(2x)/2 + C==> y = e^(x)/2 + Ce^(-x). Find C via y(0) = 1:1 = 1/2 + C ==> C = 1/2. So, the final answer is y = (e^x + e^(-x))/2. I hope this helps! tysm

Answer»

The integrating factor is e^(integral(1 dx)) = e^x.

Multiply both sides by e^x:e^x dy/dx + e^x y = e^(2x)==> (d/dx)(e^x * y) = e^(2x), by the product rule for derivatives.

Integrate both sides: e^x * y = e^(2x)/2 + C==> y = e^(x)/2 + Ce^(-x).

Find C via y(0) = 1:1 = 1/2 + C ==> C = 1/2.

So, the final answer is y = (e^x + e^(-x))/2.

I hope this helps!

tysm

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