Question

Let A be a constant. Find the general solution of y ′ − a y = 0 a). y = c e^a x b). y = − c e^a x c). y = e^a x d). y = − e^a x

Answer 1

Answer on Question #79192 – Math – Differential Equations:

Question: Let $A$ be a constant. Find the general solution of $y' - Ay = 0$.

(a). $y = c e^{Ax}$

(b). $y = -c e^{Ax}$

(c). $y = e^{Ax}$

(d). $y = -e^{Ax}$

Solution: Differential equation is given by

$$y' - Ay = 0,$$

or $\frac{dy}{dx} - Ay = 0$, $\quad [As\ y' = \frac{dy}{dx}]$

or $\frac{1}{y} \frac{dy}{dx} = A$,

or $\frac{1}{y} dy = A \, dx$ ...(1)

Now integrating both sides of equation (1) and we get

$$\ln(y) = Ax + \ln(c), \quad [\text{where } \ln c \text{ is integration constant};]$$

or $\ln\left(\frac{y}{c}\right) = A x$,

or $\frac{y}{c} = e^{Ax}$,

or $y = c e^{Ax}$.

Answer: option (a) is correct.

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