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Find the constant for which a given function satisfies a DE

This type of question, asked 3 times: 16 May 2023 S2, 17 Apr 2026 S1, 17 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    16 May 2023 S2

    124
    The function y(x)y(x) represented by x=sin⁡tx = \sin t, y=aet2+be−t2y = ae^{t\sqrt{2}} + be^{-t\sqrt{2}}, t∈(−π2,π2)t \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) satisfies the equation (1−x2)y′′−xy′=ky(1-x^2)y'' - xy' = ky, then the value of kk is

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  2. Question 2

    17 Apr 2026 S1

    87
    If y=e−mxy = e^{-mx} is a solution of the differential equation d2ydx2+4dydx+3y=0\dfrac{d^2y}{dx^2} + 4\dfrac{dy}{dx} + 3y = 0, then the values of mm are

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  3. Question 3

    17 Apr 2026 S2

    81
    The function f(x)=∫0xdt1+cos⁡tf(x) = \int_0^x\dfrac{dt}{1 + \cos t} satisfies which of the following differential equations?

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