Suppose y=y(x) be the solution curve to the differential equation (d y)/(d x)-y=2-e⁻ˣ such that lim(x → ∞) y(x) is finite. If a and b are respectively the x - and y-intercepts of the tangent to the curve at x=0, then the value of a-4 b is equal to _____________.

Solution

Maths sir · 5 steps

Solve the linear equation, then let the finite limit kill the Ceˣ term.

1. Integrating factor. The equation has the form dy/dx + Py = Q with P = -1, so the integrating factor is e^(∫ -1 dx) = e⁻ˣ. Multiplying through turns the left side into an exact derivative.

   d/dx(y e⁻ˣ) = 2e⁻ˣ - e⁻²ˣ ⇒ y e⁻ˣ = -2e⁻ˣ + 1/2e⁻²ˣ + C

2. Use the limit. Multiply by eˣ to get y = -2 + 1/2e⁻ˣ + Ceˣ. As x → ∞, Ceˣ blows up unless C = 0. So y = -2 + 1/2e⁻ˣ.
3. Tangent line. At x = 0, y(0) = -2 + 1/2 = -3/2 and y'(0) = -1/2. The tangent is y = -3/2 - 1/2x.
4. Intercepts. Put x = 0: b = -3/2. Put y = 0: x = -3, so a = -3.
5. So. a - 4b = -3 - 4(-3/2) = -3 + 6, which is 3.

The common slip: 15. Integrating -e⁻²ˣ as -1/2e⁻²ˣ instead of +1/2e⁻²ˣ gives the curve y = -2 - 1/2e⁻ˣ, intercepts a = 5 and b = -5/2, and the answer 15.

The idea to keep. A condition like a finite limit at infinity is an initial condition in disguise: any eˣ term must vanish, so its constant is 0. Keep the signs of the intercepts, they are not lengths.

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