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5.] Consider the differential equation
(dy)/(dx)=(2y)/(x). Let
y=h(x) be the particular solution to the differential equation through
(2,(5)/(e)). Find
lim_(x rarr oo)h(x).

$5$ .] Consider the differential equation $\frac{d y}{d x}=\frac{2 y}{x}$ . Let $y=h(x)$ be the particular solution to the differential equation through $\left(2, \frac{5}{e}\right)$ . Find $\lim _{x \rightarrow \infty} h(x)$ .

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Find derivatives of logarithmic functions

Full solution

Q.

5

.] Consider the differential equation

\frac{d y}{d x}=\frac{2 y}{x}

. Let

y=h(x)

be the particular solution to the differential equation through

\left(2, \frac{5}{e}\right)

. Find

\lim _{x \rightarrow \infty} h(x)

.

Recognize Separable Variables: Recognize that the differential equation is separable. Separate the variables to solve for $y$ as a function of $x$ . $\frac{dy}{y} = \frac{2dx}{x}$
Integrate Both Sides: Integrate both sides of the equation. $\newline$ $\int(\frac{1}{y})dy = \int(\frac{2}{x})dx$ $\newline$ $\ln|y| = 2\ln|x| + C$
Exponentiate to Solve: Solve for $y$ by exponentiating both sides to get rid of the natural logarithm. $\newline$ $y = e^{2\ln|x| + C}$ $\newline$ $y = |x|^2 \cdot e^C$
Drop Absolute Value: Since $y$ is always positive for this problem, we can drop the absolute value. $y = x^2 \cdot e^C$
Use Initial Condition: Use the initial condition $(2,5/e)$ to solve for $C$ . $\newline$ $\frac{5}{e} = 2^2 \cdot e^C$ $\newline$ $\frac{5}{e} = 4e^C$ $\newline$ $C = \ln\left(\frac{5}{4e}\right)$
Substitute Back for y: Substitute $C$ back into the equation for $y$ . $y = x^2 \cdot e^{\ln(\frac{5}{4e})}$ $y = x^2 \cdot \left(\frac{5}{4e}\right)$
Find Limit of $h(x)$ : Find the limit of $h(x)$ as $x$ approaches infinity. $\lim_{x \to \infty}h(x) = \lim_{x \to \infty}\left(x^2 \cdot \left(\frac{5}{4e}\right)\right)$
Limit as $x$ Approaches Infinity: Since $x^2$ grows without bound as $x$ approaches infinity, the limit is infinity. $\newline$ $\lim_{x \to \infty}h(x) = \infty$

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