Differential Equations
Equations Reducible to Linear Form
IIT-JEE 2003
Grade 12
Question:
The solution of the differential equation $\cos x \frac{dy}{dx} = y \sin x - y^2$, with $y(0) = 1$, is:
(1) $y = \frac{1}{\sin x - \cos x}$
(2) $y(\sin x + \cos x) = 1$
(3) $y = \sin x + \cos x$
(4) $y = \frac{1}{\sin x \cos x}$
Step-by-Step Solution
Key Concept: Divide throughout by $y^2$ and substitute $v = \frac{1}{y}$ to reduce this Bernoulli equation to a linear differential equation in $v$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)