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dy/dx appears inside a function or power: solve for dy/dx first

This type of question, asked 5 times: 9 May 2024 S2, 20 Apr 2025 S2, 20 Apr 2026 S1, 14 May 2026 S2, 19 May 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    9 May 2024 S2

    130
    General solution of the differential equation log⁡ ⁣(dydx)=ax+by\log\!\left(\frac{dy}{dx}\right) = ax + by is

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  2. Question 2

    20 Apr 2025 S2

    123
    The particular solution of the differential equation cos⁡(dy dx)=7,y=1\cos\left( \frac{dy}{\text{ }dx} \right)= 7,y= 1 at x=0x= 0 is

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  3. Question 3

    20 Apr 2026 S1

    85
    If y=f(x)y = f(x) is a monotonically increasing function such that (dydx)2=6−dydx\left(\dfrac{dy}{dx}\right)^2 = 6 - \dfrac{dy}{dx} and y(0)=5y(0) = 5, then y(3)=⋯y(3) = \cdots

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  4. Question 4

    14 May 2026 S2

    87
    If the function f(x)f(x) satisfying the equation [f′(x)]2+4f(x)f′(x)+f2(x)=0[f'(x)]^2 + 4f(x)f'(x) + f^2(x) = 0, then f(x)=f(x) = (where cc is constant of integration)

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  5. Question 5

    19 May 2026 S2

    87
    The general solution of edydx=xe^{\frac{dy}{dx}} = x is

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