7EDEwithY


 * DE with variable y**

Differential Equations in the form: math . \qquad \dfrac{dy}{dx}=f(y) math

Notice the variable of the derivative is **x**, but the variable of the function is **y**.

To solve,
 * first take the reciprocal,
 * then integrate with respect to **y**.

math . \qquad \dfrac{dx}{dy}=\dfrac{1}{f(y)} math . math . \qquad x = \displaystyle{\int} \dfrac{1}{f(y)} \, dy math

Then rearrange to make **y** the subject.

Example

math . \qquad \dfrac{dy}{dx}=\sqrt{4-y^2} \; \text{ given } x=\dfrac{\pi}{3}, \, y=1 math

Notice immediately that |y| __<__ 2 because the contents of the square root can't be negative

First take the reciprocal, then integrate with respect to **y**. math . \qquad \dfrac{dx}{dy}=\dfrac{1}{\sqrt{4-y^2}} \quad \textit{Notice at this point } |y| < 2 math

The variable for the derivative and the function matches (**y**) so integrate. math . \qquad x=\displaystyle{\int} \dfrac{1}{\sqrt{4-y^2}} \, dy math . math . \qquad x=\text{Sin}^{-1} \dfrac{y}{2} +c \quad \textit{General solution} math

Use the initial conditions to find **c** math \\ . \qquad \qquad \text{Given } x=\dfrac{\pi}{3}, \, y=1 \\ \\ . \qquad \qquad \dfrac{\pi}{3}=\text{Sin}^{-1} \dfrac{1}{2} +c \\ \\ . \qquad \qquad \dfrac{\pi}{3}=\dfrac{\pi}{6}+c \\ \\ . \qquad \qquad c=\dfrac{\pi}{6} math

so math . \qquad x=\text{Sin}^{-1} \dfrac{y}{2} +\dfrac{\pi}{6} math

Now transpose to make y the subject math . \qquad \text{Sin}^{-1} \dfrac{y}{2} = x-\dfrac{\pi}{6} math . math . \qquad \dfrac{y}{2} = \text{Sin} \left( x-\dfrac{\pi}{6} \right) math . math .\qquad y = 2\text{Sin} \left( x-\dfrac{\pi}{6} \right) \quad \textit{Particular solution} math This example can be solved on your calculator by entering: General Solution: math . \qquad \textbf{dSolve}(y'=\sqrt{4-y^2},x,y) math

Particular Solution: math . \qquad \textbf{dSolve}(y'=\sqrt{4-y^2},x,y,x=\frac{\pi}{3},y=1) math

Remember that the **'** symbol is on the virtual keyboard, **mth** tab, **CALC** section. The square root form and fraction form are in the **2D** tab.

Maximal Domain for this example

math \textit{Notice that, for } \text{Sin(x)} \textit{ the domain is } x\in \left[-\dfrac{\pi}{2}, \, \dfrac{\pi}{2} \right] \textit{ and the range is } y\in [-1, \, 1] math

Recall from above that **y** can't equal ±2, so math . \qquad 2\text{Sin} \left( x - \dfrac{\pi}{6} \right) \neq \pm2 \quad \textit{so} math , math , \qquad \text{Sin} \left( x - \dfrac{\pi}{6} \right) \neq \pm1 math

Thus math . \qquad -1 < \text{Sin} \left( x - \dfrac{\pi}{6} \right) < 1 math

{notice endpoints excluded because Sin can't equal 1}

math . \qquad -\dfrac{\pi}{2} < \left( x - \dfrac{\pi}{6} \right) < \dfrac{\pi}{2} math . math . \qquad -\dfrac{\pi}{2}+\dfrac{\pi}{6} < x < \dfrac{\pi}{2}+\dfrac{\pi}{6} math . math . \qquad -\dfrac{2\pi}{6} < x < \dfrac{4\pi}{6} math . math . \qquad -\dfrac{\pi}{3} < x < \dfrac{2\pi}{3} math

Thus maximal domain is: math . \qquad x \in \left( -\dfrac{\pi}{3}, \, \dfrac{2\pi}{3} \right) math

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