y₁,y₂ are different solutions of y+(ex)y=(ex+1) on [0,1]. Select the general solution, constants real.

GATE 2024 MA – Question 24

Ordinary Differential Equations · First Order Equations and Initial Value Problems · 1 mark · Multiple choice

y₁,y₂ are different solutions of $y^\prime+\sin(e^x)y=\cos(e^{x+1})$ on [0,1]. Select the general solution, constants real.

$c_1y_1+c_2y_2$
$y_1+c(y_1-y_2)$
$cy_1+(y_1-y_2)$
$c_1(y_1+y_2)+c_2(y_1-y_2)$

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Show answer and explanation

Correct answer: (B) $y_1+c(y_1-y_2)$

Explanation

Subtracting the equations shows y₁−y₂ is a nonzero homogeneous solution. A particular solution plus an arbitrary multiple of it gives the full one-parameter affine family y₁+c(y₁−y₂). **B**. Arbitrary linear combinations of two particular solutions need not retain the original forcing.

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