If the solution y(x) of the given differential equation (eʸ+1) cos x d x+eʸ sin x d y=0 passes through the point (π/2, 0), then the value of e^(y(π/6)) is equal to _________.

Solution

Maths sir · 4 steps

Divide by (eʸ+1)sin x and both sides become perfect log derivatives, so the answer drops out fast.

1. Separate variables. Divide the whole equation by (eʸ+1)sin x. You get (cos x)/(sin x) dx + eʸ/(eʸ+1) dy = 0, so each term now has only one variable.
2. Integrate. Each top is the derivative of its bottom, so each term integrates to a log: ln sin x + ln(eʸ+1) = ln c. Combining the logs gives the solution curve below.

   (eʸ+1)sin x = c

3. Find c. At (π/2, 0), e⁰ + 1 = 2 and sin π/2 = 1. So c = 2 × 1 = 2.
4. So. At x = π/6, sin π/6 = 1/2, so (eʸ+1) × 1/2 = 2. That gives eʸ + 1 = 4, so e^(y(π/6)) = 3.

The common slip: 4. Stopping at eʸ + 1 = 4 and reporting it gives 4. The question asks for eʸ, so subtract the 1.

The idea to keep. When a term looks like f'(x)/f(x), it integrates straight to ln f(x). Then join the logs into one product equation before you substitute the given point.

Answer from the NTA final answer key. Solution by PrepWiser.