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.

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- Sep 1st 2006, 07:02 PMkingkaisai2Differential equation and implicit mix- no idea
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.

- Sep 1st 2006, 08:22 PMSoroban
Hello, kingkaisai2!

This is the simplest type: Variables Separable . . .

Quote:

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: .$\displaystyle \boxed{y^2\;=\;16x^2 - 0.16}$