Let y=y₁(x) and y=y₂(x) be two distinct solutions of the differential equation (d y)/(d x)=x+y, with y₁(0)=0 and y₂(0)=1 respectively. Then, the number of points of intersection of y=y₁(x) and y=y₂(x) is

Solution

Maths sir · 4 steps

Solve the linear equation once. The two curves then differ by a pure eˣ term, and eˣ is never zero.

1. Linear form. Write the equation as dy/dx - y = x. This is linear with P = -1, so the integrating factor is e⁻ˣ.

   y e⁻ˣ = ∫ x e⁻ˣ dx = -x e⁻ˣ - e⁻ˣ + C ⇒ y = Ceˣ - x - 1

2. Fix each constant. At x = 0, y = C - 1. So y₁(0) = 0 gives C = 1 and y₁ = eˣ - x - 1. Also y₂(0) = 1 gives C = 2 and y₂ = 2eˣ - x - 1.
3. Look for crossings. The curves meet where y₂ - y₁ = 0. But y₂ - y₁ = eˣ, and eˣ > 0 for every real x, so this equation has no solution.
4. So. The curves never intersect. The number of points of intersection is 0, option (a).

The idea to keep. For a linear equation, two solutions differ only through the constant term C e^(-∫ P dx). An exponential is never zero, so different solution curves never meet.

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