Formation of Differential Equations PYQs
Practice Class 12 CBSE Board Previous Year Questions (2008-2026)
Q1
2008 Board
00:00
Form the differential equation representing the family of curves $y = Ae^{2x} + Be^{-2x}$. 3 Marks
Step 1: Differentiate Once
$y = Ae^{2x} + Be^{-2x}$
$\frac{dy}{dx} = 2Ae^{2x} - 2Be^{-2x}$
Step 2: Differentiate Twice
$\frac{d^2y}{dx^2} = 4Ae^{2x} + 4Be^{-2x}$
Step 3: Eliminate Arbitrary Constants
$\frac{d^2y}{dx^2} = 4(Ae^{2x} + Be^{-2x})$
$\frac{d^2y}{dx^2} = 4y \Rightarrow \frac{d^2y}{dx^2} - 4y = 0$.
d²y/dx² - 4y = 0
Q2
2009 Board
00:00
Form the differential equation of the family of curves $y = A\cos x + B\sin x$. 3 Marks
Step 1: Differentiate Once
$\frac{dy}{dx} = -A\sin x + B\cos x$
Step 2: Differentiate Twice
$\frac{d^2y}{dx^2} = -A\cos x - B\sin x$
Step 3: Eliminate Constants
$\frac{d^2y}{dx^2} = -(A\cos x + B\sin x) = -y$
$\frac{d^2y}{dx^2} + y = 0$.
d²y/dx² + y = 0
Q3
2011 Board
00:00
Form the differential equation of the family of parabolas $y^2 = 4a(x+b)$. 4 Marks
Step 1: Differentiate with respect to x
$2y \frac{dy}{dx} = 4a \Rightarrow y \frac{dy}{dx} = 2a$.
Step 2: Differentiate again to eliminate 'a'
Differentiating $y \frac{dy}{dx} = 2a$ gives:
$y \frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 = 0$.
y(d²y/dx²) + (dy/dx)² = 0
Q4
2010 Board
00:00
Form the differential equation representing $y = Ae^x + Be^{-x}$. 3 Marks
$y = Ae^x + Be^{-x}$
$\frac{dy}{dx} = Ae^x - Be^{-x}$
$\frac{d^2y}{dx^2} = Ae^x + Be^{-x} = y$
$\frac{d^2y}{dx^2} - y = 0$.
d²y/dx² - y = 0
Q5
2013 Board
00:00
Form the differential equation representing $y = A\sin 2x + B\cos 2x$. 3 Marks
$\frac{dy}{dx} = 2A\cos 2x - 2B\sin 2x$
$\frac{d^2y}{dx^2} = -4A\sin 2x - 4B\cos 2x = -4y$
$\frac{d^2y}{dx^2} + 4y = 0$.
d²y/dx² + 4y = 0
Q6
2014 Board
00:00
Form the differential equation representing $y = Ae^{3x} + Be^{-3x}$. 3 Marks
$\frac{d^2y}{dx^2} = 9Ae^{3x} + 9Be^{-3x} = 9y$
$\frac{d^2y}{dx^2} - 9y = 0$.
d²y/dx² - 9y = 0
Q7
2016 Board
00:00
Form the differential equation representing $y = A\cos 3x + B\sin 3x$. 3 Marks
$\frac{d^2y}{dx^2} = -9y \Rightarrow \frac{d^2y}{dx^2} + 9y = 0$.
d²y/dx² + 9y = 0
Q8
2018 Board
00:00
Form the differential equation representing $y = Ae^{ax}$. 2 Marks
$\frac{dy}{dx} = aAe^{ax} = ay$
$\frac{dy}{dx} - ay = 0$.
dy/dx - ay = 0
Q9
2020 Board
00:00
Form the differential equation representing $y = A\sin x + B\cos x$. 3 Marks
$\frac{dy}{dx} = A\cos x - B\sin x$
$\frac{d^2y}{dx^2} = -A\sin x - B\cos x = -y$
$\frac{d^2y}{dx^2} + y = 0$.
d²y/dx² + y = 0
Q10
2022 Board
00:00
Form the differential equation representing $y = Ae^{2x} + Be^{-2x}$. 3 Marks
$\frac{d^2y}{dx^2} = 4y \Rightarrow \frac{d^2y}{dx^2} - 4y = 0$.
d²y/dx² - 4y = 0
Q11
2024 Board
00:00
Form the differential equation representing $y = A\cos x + B\sin x$. 3 Marks
$\frac{d^2y}{dx^2} + y = 0$.
d²y/dx² + y = 0
Q12
2026 Board
00:00
Form the differential equation representing $y = Ae^{3x} + Be^{-3x}$. 3 Marks
$\frac{d^2y}{dx^2} = 9y \Rightarrow \frac{d^2y}{dx^2} - 9y = 0$.
d²y/dx² - 9y = 0