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The differential equation presented is solved by separating variables and integrating both sides, leading to the expression involving arcsin functions. The solution is expressed in terms of sine and cosine using the compound angle formula and the Pythagorean identity. Constants A and B are introduced, with a suggestion that they are not independent, leading to a refined expression for y. The final form of the solution is presented as y = x√(1 - B²) + B√(1 - x²). This approach highlights the relationships between the variables and constants in the context of the equation.
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Solve the Differential Equation $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} = \sqrt{ \frac{1 - y^2}{1 - x^2} } \end{align*}$

$\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \sqrt{ \frac{1 - y^2}{1 - x^2} } \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{\sqrt{ 1 - y^2 }}{\sqrt{1 - x^2} } \\ \frac{1}{\sqrt{1 - y^2}} \, \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{1}{\sqrt{1 - x^2} } \\ \int{ \frac{1}{\sqrt{1 - y^2}}\, \frac{\mathrm{d}y}{\mathrm{d}x} \, \mathrm{d}x} &= \int{\frac{1}{\sqrt{1 - x^2}} \, \mathrm{d}x} \\ \int{ \frac{1}{\sqrt{1 - y^2}} \, \mathrm{d}y} &= \arcsin{(x)} + C_1 \\ \arcsin{(y)} + C_2 &= \arcsin{(x)} + C_1 \\ \arcsin{(y)} &= \arcsin{(x)} + C \textrm{ where } C = C_1 - C_2 \\ y &= \sin{ \left[ \arcsin{(x)} + C \right] } \\ y &= \sin{ \left[ \arcsin{(x)} \right] } \cos{(C)} + \cos{ \left[ \arcsin{(x)} \right] } \sin{(C)} \\ y &= A\sin{ \left[ \arcsin{(x)} \right] } + B\,\cos{ \left[ \arcsin{(x)} \right] } \textrm{ where } A = \cos{(C)} \textrm{ and } B = \sin{(C)} \\ y &= A\,x + B\,\sqrt{ 1 - \left\{ \sin{ \left[ \arcsin{(x)} \right] } \right\} ^2 } \\ y &= A\,x + B\,\sqrt{1 - x^2 } \end{align*}$

Make note of my use of the Compound Angle Formula for sine, and the Pythagorean Identity.
 
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I think you should note that $A$ and $B$ are not independent constants... it might be better to write:

$y = x\sqrt{1 - B^2} + B\sqrt{1 - x^2}$
 
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