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If
(dy)/(dx)=ky where
k is a constant, and
y(0)=100, then
y(3)
A.
100e^(k)

(dy)/(dx)=ky rarr int(1)/(y)dy+(kdx
B.
300e^(k)

ln(y)=kx+c
C.
100e^(3k)

ln()=
D.
300e^(3k)

If $\frac{d y}{d x}=k y$ where $k$ is a constant, and $y(0)=100$ , then $y(3)$ $\newline$ A. $100 e^{k}$ $\newline$ $\frac{d y}{d x}=k y \rightarrow \int \frac{1}{y} d y+(k d x$ $\newline$ B. $300 e^{k}$ $\newline$ $\ln (y)=k x+c$ $\newline$ C. $100 e^{3 k}$ $\newline$ $\ln ()=$ $\newline$ D. $300 e^{3 k}$

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Transformations of absolute value functions

Full solution

Q. If

\frac{d y}{d x}=k y

where

k

is a constant, and

y(0)=100

, then

y(3)

\newline

A.

100 e^{k}

\newline

\frac{d y}{d x}=k y \rightarrow \int \frac{1}{y} d y+(k d x

\newline

B.

300 e^{k}

\newline

\ln (y)=k x+c

\newline

C.

100 e^{3 k}

\newline

\ln ()=

\newline

D.

300 e^{3 k}

Separate and Integrate: Solve the differential equation $\frac{dy}{dx} = ky$ . To solve this first-order linear differential equation, we separate variables and integrate both sides.
Perform Integration: Separate variables and integrate. $\newline$ We write the equation as $(\frac{1}{y})dy = kdx$ and integrate both sides. $\newline$ $\int(\frac{1}{y})dy = \int kdx$
Solve for $y$ : Perform the integration. $\newline$ The integral of $\frac{1}{y}$ dy is $\ln|y|$ , and the integral of $kdx$ is $kx$ . So we have: $\newline$ $\ln|y| = kx + C$ , where $C$ is the constant of integration.
Determine Constant $C$ : Solve for $y$ . $\newline$ To solve for $y$ , we exponentiate both sides of the equation to get rid of the natural logarithm. $\newline$ $e^{\ln|y|} = e^{kx + C}$ $\newline$ $|y| = e^{kx} \cdot e^C$ $\newline$ Since $y$ is positive (given $y(0) = 100$ ), we can drop the absolute value: $\newline$ $y = e^{kx} \cdot e^C$
Substitute for $C$ : Determine the constant $C$ using the initial condition $y(0) = 100$ . Substitute $x = 0$ and $y = 100$ into the equation $y = e^{kx} \cdot e^C$ to find $C$ . $100 = e^{k\cdot 0} \cdot e^C$ $100 = e^0 \cdot e^C$ $100 = 1 \cdot e^C$ $C$ $0$
Substitute $C$ into $y$ : Substitute $C$ back into the equation for $y$ .
$y = e^{kx} \cdot e^{\ln(100)}$
$y = e^{kx} \cdot 100$
Find $y(3)$ : Find $y(3)$ . $\newline$ Substitute $x = 3$ into the equation $y = e^{kx} \times 100$ to find $y(3)$ . $\newline$ $y(3) = e^{k\times 3} \times 100$ $\newline$ $y(3) = 100e^{3k}$

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