GATE 2025 MA – Question 37

Ordinary Differential Equations · Higher Order Equations and Systems · 2 marks · Multiple choice

x²y″+7xy′+9y=x⁻³ ln x for x>0, with y(1)=y′(1)=0. Find y(e).

$e^{-3}/3$
$e^{-3}/6$
$2e^{-3}/3$
$e^{-3}/2$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) $e^{-3}/6$

Explanation

Put t=lnx. The Euler operator becomes $(D_t+3)^2y=e^{-3t}t$. Write y=e⁻³ᵗv(t), giving v″=t. Initial conditions imply v(0)=v′(0)=0, so v=t³/6. At x=e,t=1, **y=e⁻³/6, B**.

More Higher Order Equations and Systems questions

GATE 2025 MA Q38 – Independent solutions of (1+x²)y″−xy′+cos²x y=0 satisfy…
GATE 2024 MA Q45 – A is a nondiagonalizable real 2×2 matrix with real eigenvalue λ and eigenvector v. Choose…
GATE 2024 MA Q63 – The BVP x²y″−2xy′+2y=0 on [1,2], y(1)−y′(1)=1, y(2)−ky′(2)=4, has infinitely many…
GATE 2023 MA Q33 – Solve y″+4y=t for 0≤t≤2 and =2 for t>2, with y(0)=y′(0)=0. If α=y(π/2), find (4/π)α to two…
GATE 2023 MA Q41 – Let φ,ψ be independent solutions of y″+(2−cos x)y=0 on ℝ. Suppose α<β,φ(α)=φ(β)=0 and φ≠0…
GATE 2023 MA Q49 – For x′=y,y′=x with x(0)=α,y(0)=β, which choices give x(t) + y(t) →0 as t→∞?

All Higher Order Equations and Systems questions →