Let y = y(x) be the solution of the differential equation dy = e αx + y dx; α ∈ N. If y(logₑ2) = logₑ2 and y(0) = logₑ( 1/2 ), then the value of α is equal to _____________.

Solution

Maths sir · 4 steps

Split e^(α x + y) into a product, separate, then let the two points fix C and α.

1. Separate the variables. Since e^(α x + y) = e^(α x) eʸ, divide both sides by eʸ to get e⁻ʸ dy = e^(α x) dx. Integrating both sides gives the general solution.

   -e⁻ʸ = e^(α x)/α + C

2. Use y(0). At x = 0, y = ln 1/2, so e⁻ʸ = 2. Then -2 = 1/α + C, which gives C = -2 - 1/α.
3. Use y(ln 2). At x = ln 2, y = ln 2, so e⁻ʸ = 1/2 and e^(α x) = 2^α. Then -1/2 = 2^α/α - 2 - 1/α, which simplifies to 3/2 = (2^α - 1)/α, that is 2^(α + 1) = 3α + 2.
4. So. Try natural numbers. For α = 1: 4 ≠ 5. For α = 2: 8 = 8. For α ≥ 3 the left side grows much faster, so α = 2.

The idea to keep. When the exponent is a sum like eᵃˣ⁺ʸ, split it into a product and separate. Two conditions mean two unknowns: one point gives C, the other gives the parameter.

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