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Step 4 Now we integrate both sides of the equation we have found with the integrating factor. $\int \frac{d}{dx}[e^{-2x}y] dx = \int x^2e^{-2x} + 3e^{-2x} dx$ Note that the left side of the equation is the integral of the derivative of $e^{-2x}y$. Therefore, up to a constant of integration, the left side reduces as follows. $\int \frac{d}{dx}[e^{-2x}y] dx = e^{-2x}y$ The integration on the right side of the equation requires integration by parts. $\int x^2e^{-2x} + 3e^{-2x} dx = (-\frac{x^2e^{-2x}}{2} - \frac{xe^{-2x}}{2} - \frac{e^{-2x}}{4}) - ( )e^{-2x} + C$ $= e^{-2x}(-\frac{x^2}{2} - \frac{x}{2} - \frac{1}{4}) - ( ) + C$ $= e^{-2x}(-\frac{x^2}{2} - \frac{x}{2} - ( )) + C$

          Step 4
Now we integrate both sides of the equation we have found with the integrating factor.
$\int \frac{d}{dx}[e^{-2x}y] dx = \int x^2e^{-2x} + 3e^{-2x} dx$
Note that the left side of the equation is the integral of the derivative of $e^{-2x}y$. Therefore, up to a constant of integration, the left side reduces as follows.
$\int \frac{d}{dx}[e^{-2x}y] dx = e^{-2x}y$
The integration on the right side of the equation requires integration by parts.
$\int x^2e^{-2x} + 3e^{-2x} dx = (-\frac{x^2e^{-2x}}{2} - \frac{xe^{-2x}}{2} - \frac{e^{-2x}}{4}) - ( )e^{-2x} + C$
$= e^{-2x}(-\frac{x^2}{2} - \frac{x}{2} - \frac{1}{4}) - ( ) + C$
$= e^{-2x}(-\frac{x^2}{2} - \frac{x}{2} - ( )) + C$

Step 4
Now we integrate both sides of the equation we have found with the integrating factor.
∫(d)/(dx)[e^-2xy] dx = ∫ x^2e^-2x + 3e^-2x dx
Note that the left side of the equation is the integral of the derivative of e^-2xy. Therefore, up to a constant of integration, the left side reduces as follows.
∫(d)/(dx)[e^-2xy] dx = e^-2xy
The integration on the right side of the equation requires integration by parts.
∫ x^2e^-2x + 3e^-2x dx = (-(x^2e^-2x)/(2) - (xe^-2x)/(2) - (e^-2x)/(4)) - ( )e^-2x + C
= e^-2x(-(x^2)/(2) - (x)/(2) - (1)/(4)) - ( ) + C
= e^-2x(-(x^2)/(2) - (x)/(2) - ( )) + C

Added by Amanda B.

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