Use integration to find a general solution of the differential equation

Question

Quizplus · Accepted Answer

The differential equation can be written as: 
$$\frac{dy}{dx} + 2y = 4xe^{-x}$$
Multiplying both sides by $e^{2x}$, we get: 
$$e^{2x}\frac{dy}{dx} + 2ye^{2x} = 4xe^{x}$$
This is the product rule of $(ye^{2x})$, so we can write: 
$$\frac{d}{dx}(ye^{2x}) = 4xe^x$$
Integrating both sides with respect to $x$, we get: 
$$ye^{2x} = \int 4xe^x dx$$
Using integration by parts with $u = x$ and $dv = e^x$, we get: 
$$ye^{2x} = 4xe^x - \int 4e^x dx = 4xe^x - 4e^x + C$$
where $C$ is the constant of integration. 
Dividing both sides by $e^{2x}$, we get the general solution: 
$$y = \frac{4x}{e^x} - \frac{4}{e^{2x}} + Ce^{-2x}$$
which is option B. The differential equation is:
$$\frac{dy}{dx} = (1+y^2)e^x$$

We can separate variables and integrate both sides:
$$\int \frac{dy}{1+y^2} = \int e^x dx$$
Using substitution $y = 	an u$ and $dy = \sec^2 u du$, we get:
$$\int \frac{dy}{1+y^2} = \int \cos^{-2} u du$$
$$= \int \sec^2 u du = 	an u = 	an^{-1} y + C_1$$
And
$$\int e^x dx = e^x + C_2$$
Therefore, the general solution is:
$$	an^{-1} y  = e^x + C$$
where $C = C_1+C_2$ is the constant of integration.

Simplifying this expression we get, 
$$y = 	an (e^x+C)$$

So, the correct choice is B.

Use Integration to Find a General Solution of the Differential