Subtopic: Higher order linear ODEs with constant coefficients
Question Statement:
The complete solution for the differential equation $\frac{d^2y}{dx^2} + 2\frac{dy}{dx} - 3y = 0$ for $y(0) = 0$ and $\left(\frac{dy}{dx}\right)_{x=0} = 8$ is ______
Options:
(A) $2(e^x - e^{3x})$
(B) $2(e^x - e^{-3x})$
(C) $2(e^x + e^{3x})$
(D) $2(e^x + e^{-3x})$
Official Answer Key:
B
Step-by-Step Solution & Explanation
Step 1 — auxiliary equation:
m² + 2m − 3 = 0
(m + 3)(m − 1) = 0 ⇒ m = 1, −3
Step 2 — general solution:
y = A eˣ + B e⁻³ˣ
Step 3 — apply the initial conditions.
y(0) = 0: A + B = 0 ⇒ B = −A
y′ = A eˣ − 3B e⁻³ˣ ; y′(0) = 8: A − 3B = 8
Substituting B = −A: A + 3A = 8 ⇒ A = 2, B = −2
Step 4 — complete solution:
y = 2eˣ − 2e⁻³ˣ = 2(eˣ − e⁻³ˣ)
Check: the roots are +1 and −3, so the exponents must be x and −3x — this alone eliminates (A) and (C).
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