# Write and solve the differential equation that models the verbal statement. 14) The rate of change of y with respect to x varies jointly as x and L - y ... I don't know what it means by...

Write and solve the differential equation that models the verbal statement.

**14)** The rate of change of *y* with respect to *x *varies jointly as *x* and *L - y*

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I don't know what it means by "varies jointly as *x *and *L - y.*"

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If y varies directly with x we can write an equation y=kx that describes the relationship between x and y. k is called the constant of proportionality.

If z varies jointly with y and x, then z varies directly with both x and y and we write z=kxy.

Here the rate of change of y with respect to x (or `(dy)/(dx) ` ), varies jointly with x and L-y. (We assume L is a constant.)

Thus `(dy)/(dx)=kx(L-y) ` or `dy=kx(L-y)dx ` which is a first order differential equation. Use the separation of variables technique:

`dy=kx(L-y)dx `

`(dy)/(L-y)=kxdx `

Integrating both sides we get:

`-ln(L-y)=(kx^2)/2+C ` where ln is the natural logarithm and C is a constant.

Exponentiating both sides with base e we get:

`L-y=e^((-kx^2)/2+C) ` or `L-y=C_1e^((-kx^2)/2) `

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Then `y=Ce^((-kx^2)/2)+L `

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