Differential Equations
Differential Equation
nta_abhyas_2025
Grade 12
Question:
The solution of the differential equation $\sin(x + y)dy = dx$ is
$y + \tan(x + y) - \sec(x + y) = c$
$y - \tan(x + y) - \sec(x + y) = c$
$y + \tan(x + y) + \sec(x + y) = c$
$y - \tan(x + y) + \sec(x + y) = c$
Step-by-Step Solution
Key Concept: Use substitution $x + y = v$ to convert a differential equation into a separable or integrable form.
Let $x + y = v$, so $\frac{dv}{dx} = 1 + \frac{dy}{dx}$. The equation becomes $\frac{dv}{dx} - 1 = \tan v + \sec v$, or $\frac{dv}{dx} = 1 + \tan v + \sec v$. Rearranging: $\frac{dv}{1+\tan v + \sec v} = dx$. Integrating using $\int(\sec v - \tan v)\,dv = \sec v - \tan v + C$, we obtain $x = v - \tan(v/2) + \sec(x+y) + c$, which simplifies to $x = y - \tan(x+y) + \sec(x+y) + c$.
Correct Answer: c