Find the particular solution of the differential equation that satisfies the initial condition

Question

Find the particular solution of the differential equation  that satisfies the initial condition

Quizplus · Accepted Answer

We can first solve the differential equation with the given initial condition:
$$\frac{dy}{dx} = 3y, \quad y(0)=5$$
Separating variables and integrating:
$$\int \frac{1}{y} dy = \int 3 dx$$
$$\ln|y| = 3x + C$$
$$y = Ce^{3x}$$
Using the initial condition to solve for C:
$$y(0) = C = 5$$
Therefore, the particular solution is:
$$y = 5e^{3x}$$
Thus, the correct choice is B. The differential equation can be written as
$$\frac{dy}{dx} + 2y = 3e^{-2x}$$
The integrating factor is $e^{2x}$, so we multiply both sides by it:
$$e^{2x}\frac{dy}{dx} + 2ye^{2x} = 3$$
$$\frac{d}{dx}(ye^{2x}) = 3$$
Integrating both sides with respect to $x$ gives
$$ye^{2x} = 3x+C$$
where $C$ is an arbitrary constant. Using the initial condition $y(0)=1$, we have
$$e^0 = 3\cdot0+C \Rightarrow C=1$$
So the particular solution that satisfies the initial condition is given by
$$y=\frac{1}{e^{2x}}(3x+1)$$
which corresponds to choice B.

Find the Particular Solution of the Differential Equation That