Question:
(c) Solve the differential equation \( (\tan^{-1} y - x) dy = (1 + y^2) dx \):
(c) Solve the differential equation \( (\tan^{-1} y - x) dy = (1 + y^2) dx \):
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Substitution is a powerful tool for solving first-order differential equations.
Updated On: Mar 1, 2025
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Solution and Explanation
Rewriting the equation:
\[
\frac{dy}{dx} = \frac{1 + y^2}{\tan^{-1} y - x}.
\]
Use substitution \( z = \tan^{-1} y - x \), then differentiate and solve. The solution is:
\[
z = C, \quad \text{where } z = \tan^{-1} y - x.
\]
Thus:
\[
\tan^{-1} y - x = C.
\]
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