Question

Choose the right solution of the differential equation \[\dfrac{{dy}}{{dx}} + \sqrt {\left( {\dfrac{{1 - {y^2}}}{{1 - {x^2}}}} \right)} = 0\] from the following options.
A. \[{\tan ^{ - 1}}x + {\cot ^{ - 1}}x = c\]
B. \[{\sin ^{ - 1}}x + {\sin ^{ - 1}}y = c\]
C. \[{\sec ^{ - 1}}x + \cos e{c^{ - 1}}x = c\]
D. None of these

Answer 1

Hint: The solution of the differential equation can be found using the variable separation method. There are two variables x and y. Separate x containing terms and y containing terms. Then integrate the equation to find a general solution.

Formula Used: \[\int {\dfrac{{dx}}{{\sqrt {1 - {x^2}} }} = {{\sin }^{ - 1}}x + c} \]

Complete step by step solution: The given differential equation is \[\dfrac{{dy}}{{dx}} + \sqrt {\left( {\dfrac{{1 - {y^2}}}{{1 - {x^2}}}} \right)} = 0\].
First simplify the equation by subtracting \[\sqrt {\left( {\dfrac{{1 - {y^2}}}{{1 - {x^2}}}} \right)} \] both sides of the equation.
\[\dfrac{{dy}}{{dx}} = - \sqrt {\left( {\dfrac{{1 - {y^2}}}{{1 - {x^2}}}} \right)} \]
Further simplify the equation as follows.
\[\dfrac{{dy}}{{dx}} = - \dfrac{{\sqrt {1 - {y^2}} }}{{\sqrt {1 - {x^2}} }}\]
Multiply by \[dx\]on both sides of the equation.
\[dy = - \dfrac{{\sqrt {1 - {y^2}} }}{{\sqrt {1 - {x^2}} }}dx\]
Multiply by \[\dfrac{1}{{\sqrt {1 - {y^2}} }}\] on both sides of the equation.
\[\dfrac{{dy}}{{\sqrt {1 - {y^2}} }} = - \dfrac{{dx}}{{\sqrt {1 - {x^2}} }}\]
Integrate both sides of the equation.
 \[\begin{array}{l}\int {\dfrac{{dy}}{{\sqrt {1 - {y^2}} }}} = - \int {\dfrac{{dx}}{{\sqrt {1 - {x^2}} }}} \\{\sin ^{ - 1}}y = - {\sin ^{ - 1}}x - c\end{array}\]
Add \[{\sin ^{ - 1}}x\]on both sides of the equation.
\[{\sin ^{ - 1}}y + {\sin ^{ - 1}}x = c\]
So, \[{\sin ^{ - 1}}y + {\sin ^{ - 1}}x = c\] is the general solution of the equation.

Option ‘B’ is correct

Note: The common mistake happens here is instead of using \[{\sin ^{ - 1}}x\]formula while performing the integration students use to solve the term \[\dfrac{1}{{\sqrt {1 - {y^2}} }}\] for integration.