a. Two quantities x and y are related to each other by the differential equation ydy/dx=-16x. Solve this equation to get an implicit equation of the solution curve for which y=0 when x=0.1.
Hello, kingkaisai2!
This is the simplest type: Variables Separable . . .
a. Two quantities $\displaystyle x$ and $\displaystyle y$ are related to each other
. . by the differential equation: $\displaystyle y\,\frac{dy}{dx} = -16x$
Solve this equation to get an implicit equation of the solution curve
. . for which $\displaystyle y=0$ when $\displaystyle x=0.1.$
We have: .$\displaystyle y\,dy \:=\:16x\,dx$
Integrate: .$\displaystyle \int y\,dy \:=\:\int 16x\,dx\quad\Rightarrow\quad \frac{1}{2}y^2\:=\:8x^2$$\displaystyle + c\quad\Rightarrow\quad y^2\:=\:16x^2 + C$
When $\displaystyle x = 0.1,\;y = 0:\;\;0^2 \:=\:16(0.1)^2 + C\quad\Rightarrow\quad C = -0.16$
Therefore: .$\displaystyle \boxed{y^2\;=\;16x^2 - 0.16}$