Let a smooth curve y=f(x) have slope of tangent directly proportional to -yx.

Question

Let a smooth curve $y=f(x)$ have slope of tangent directly proportional to $-\dfrac{y}{x}$. If it passes through $(1,2)$ and $(8,1)$, then $y\left(\dfrac18\right)$ is:

Accepted Answer

$$\frac{dy}{dx}=-\frac{ky}{x} \implies \ln|y|=-k\ln|x|+C.$$

Using $(1,2)$: $C=\ln2$. Using $(8,1)$: $0=-k\ln8+\ln2 \implies k=\dfrac13$.

At $x=1/8$: $\ln|y| = -\dfrac13\ln\dfrac18+\ln2 = \ln1+\ln2=\ln4 \implies |y|=4$.

Solutions · 1