Linear Differential Equations PYQs
Practice Class 12 CBSE Board Previous Year Questions (2008-2026)
Q1
2009 Board
00:00
Solve the differential equation $\frac{dy}{dx} + y = e^{-x}$. 3 Marks
Step 1: Identify P and Q
This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$.
Here, $P = 1$ and $Q = e^{-x}$.
Step 2: Find Integrating Factor (I.F.)
$I.F. = e^{\int P \, dx} = e^{\int 1 \, dx} = e^x$.
Step 3: Solve the Equation
$y(I.F.) = \int Q(I.F.) \, dx + C$
$y(e^x) = \int e^{-x} \cdot e^x \, dx + C$
$ye^x = \int 1 \, dx + C = x + C$.
Final solution: $y = (x+C)e^{-x}$.
y = (x+C)e⁻ˣ
Q2
2011 Board
00:00
Solve the differential equation $x\frac{dy}{dx} + y = x^2 \sin x$. 4 Marks
Step 1: Divide by x to get standard form
$\frac{dy}{dx} + \frac{1}{x}y = x \sin x$.
Here, $P = 1/x$ and $Q = x \sin x$.
Step 2: Find I.F.
$I.F. = e^{\int (1/x) \, dx} = e^{\log x} = x$.
Step 3: Solve
$yx = \int (x \sin x) \cdot x \, dx + C = \int x^2 \sin x \, dx + C$.
Using integration by parts: $yx = -x^2 \cos x + 2x \sin x + 2 \cos x + C$.
yx = -x² cos x + 2x sin x + 2 cos x + C
Q3
2014 Board
00:00
Solve the differential equation $\frac{dy}{dx} + y \tan x = \sin x$. 4 Marks
Step 1: Identify P and Q
$P = \tan x, Q = \sin x$.
Step 2: Find I.F.
$I.F. = e^{\int \tan x \, dx} = e^{\log|\sec x|} = \sec x$.
Step 3: Solve
$y \sec x = \int \sin x \cdot \sec x \, dx + C = \int \tan x \, dx + C$.
$y \sec x = \log|\sec x| + C$.
y sec x = log|sec x| + C
Q4
2026 Board
00:00
Solve the differential equation $\frac{dy}{dx} + y = e^x$. 3 Marks
Step 1: Identify P and Q
$P = 1, Q = e^x$.
Step 2: Find I.F.
$I.F. = e^{\int 1 \, dx} = e^x$.
Step 3: Solve
$y e^x = \int e^x \cdot e^x \, dx + C = \int e^{2x} \, dx + C$.
$y e^x = \frac{e^{2x}}{2} + C \Rightarrow y = \frac{e^x}{2} + Ce^{-x}$.
y = eˣ/2 + Ce⁻ˣ
Q5
2024 Board
00:00
Solve the differential equation $x \frac{dy}{dx} - y = x^2$. 3 Marks
Step 1: Standard Form
$\frac{dy}{dx} - \frac{1}{x}y = x$.
Step 2: Find I.F.
$I.F. = e^{\int (-1/x) \, dx} = e^{-\log x} = 1/x$.
Step 3: Solve
$y(1/x) = \int x(1/x) \, dx + C = \int 1 \, dx + C = x + C$.
$y = x^2 + Cx$.
y = x² + Cx
Q6
2023 Board
00:00
Solve the differential equation $x \frac{dy}{dx} - y = x^2 e^x$. 4 Marks
Step 1: Standard Form
$\frac{dy}{dx} - \frac{1}{x}y = x e^x$.
Step 2: Find I.F.
$I.F. = 1/x$.
Step 3: Solve
$y(1/x) = \int x e^x (1/x) \, dx + C = \int e^x \, dx + C = e^x + C$.
$y = x e^x + Cx$.
y = x eˣ + Cx
Q7
2022 Board
00:00
Solve the differential equation $\frac{dy}{dx} + 2y = e^x$. 3 Marks
Step 1: Find I.F.
$P = 2 \Rightarrow I.F. = e^{2x}$.
Step 2: Solve
$y e^{2x} = \int e^x \cdot e^{2x} \, dx + C = \int e^{3x} \, dx + C$.
$y e^{2x} = \frac{e^{3x}}{3} + C \Rightarrow y = \frac{e^x}{3} + Ce^{-2x}$.
y = eˣ/3 + Ce⁻²ˣ
Q8
2019 Board
00:00
Solve the differential equation $x \frac{dy}{dx} - 2y = x^2$. 4 Marks
Step 1: Standard Form
$\frac{dy}{dx} - \frac{2}{x}y = x$.
Step 2: Find I.F.
$I.F. = e^{\int (-2/x) \, dx} = e^{-2\log x} = 1/x^2$.
Step 3: Solve
$y(1/x^2) = \int x(1/x^2) \, dx + C = \int 1/x \, dx + C = \log|x| + C$.
$y = x^2 \log|x| + Cx^2$.
y = x² log|x| + Cx²
Q9
2008 Board
00:00
Solve the differential equation $\frac{dy}{dx} - y = x$. 3 Marks
Step 1: Find I.F.
$P = -1 \Rightarrow I.F. = e^{-x}$.
Step 2: Solve
$y e^{-x} = \int x e^{-x} \, dx + C$.
Using integration by parts: $y e^{-x} = -x e^{-x} - e^{-x} + C$.
$y = -x - 1 + Ce^x$.
y = -x - 1 + Ceˣ
Q10
2023 Board
00:00
Solve the differential equation $\frac{dy}{dx} = x + y$. 3 Marks
$\frac{dy}{dx} - y = x$.
I.F. $= e^{-x}$.
$y e^{-x} = \int x e^{-x} \, dx + C = -x e^{-x} - e^{-x} + C$.
$y = -x - 1 + Ce^x$.
y = -x - 1 + Ceˣ
Q11
2021 Board
00:00
Solve the differential equation $x \frac{dy}{dx} + y = x^3$. 3 Marks
$\frac{dy}{dx} + \frac{1}{x}y = x^2$.
I.F. $= x$.
$yx = \int x^3 \, dx + C = \frac{x^4}{4} + C$.
$y = \frac{x^3}{4} + \frac{C}{x}$.
y = x³/4 + C/x
Q12
2020 Board
00:00
Solve the differential equation $\frac{dy}{dx} + \frac{y}{x} = x$. 2 Marks
I.F. $= x$.
$yx = \int x^2 \, dx + C = \frac{x^3}{3} + C$.
$y = \frac{x^2}{3} + \frac{C}{x}$.
y = x²/3 + C/x
Q13
2015 Board
00:00
Solve the differential equation $\frac{dy}{dx} + 2xy = x$. 3 Marks
I.F. $= e^{x^2}$.
$y e^{x^2} = \int x e^{x^2} \, dx + C = \frac{1}{2} e^{x^2} + C$.
$y = \frac{1}{2} + Ce^{-x^2}$.
y = 1/2 + Ce⁻ˣ²
Q14
2012 Board
00:00
Solve the differential equation $\frac{dy}{dx} + 2y = e^{-x}$. 3 Marks
I.F. $= e^{2x}$.
$y e^{2x} = \int e^x \, dx + C = e^x + C$.
$y = e^{-x} + Ce^{-2x}$.
y = e⁻ˣ + Ce⁻²ˣ
Q15
2026 Board
00:00
Solve the differential equation $x \frac{dy}{dx} - y = x^2$. 3 Marks
$\frac{dy}{dx} - \frac{1}{x}y = x$.
I.F. $= 1/x$.
$y/x = x + C \Rightarrow y = x^2 + Cx$.
y = x² + Cx
Q16
2025 Board
00:00
Solve the differential equation $x \frac{dy}{dx} + 2y = x^2$. 3 Marks
$\frac{dy}{dx} + \frac{2}{x}y = x$.
I.F. $= x^2$.
$yx^2 = \int x^3 \, dx + C = x^4/4 + C$.
$y = x^2/4 + C/x^2$.
y = x²/4 + C/x²
Q17
2020 Board
00:00
Solve the differential equation $\frac{dy}{dx} + \frac{y}{x} = x$. 2 Marks
I.F. $= x$.
$yx = x^3/3 + C$.
y = x²/3 + C/x
Q18
2018 Board
00:00
Solve the differential equation $x \frac{dy}{dx} + y = x e^x$. 4 Marks
$\frac{dy}{dx} + \frac{1}{x}y = e^x$.
I.F. $= x$.
$yx = \int x e^x \, dx + C = (x-1)e^x + C$.
yx = (x-1)eˣ + C
Q19
2017 Board
00:00
Solve the differential equation $\frac{dy}{dx} + \frac{y}{x} = \frac{x}{\log x}$. 4 Marks
I.F. $= x$.
$yx = \int \frac{x^2}{\log x} \, dx + C$.
Solve using appropriate substitution or leave in integral form if complex.
yx = ∫ (x²/log x) dx + C
Q20
2016 Board
00:00
Solve the differential equation $x \frac{dy}{dx} + 2y = x^2$. 4 Marks
I.F. $= x^2$.
$yx^2 = x^4/4 + C$.
y = x²/4 + C/x²
Q21
2014 Board
00:00
Solve the differential equation $x \frac{dy}{dx} - y = x^2$. 3 Marks
I.F. $= 1/x$.
$y/x = x + C$.
y = x² + Cx
Q22
2013 Board
00:00
Solve the differential equation $\frac{dy}{dx} + \frac{y}{x} = \sin x$. 4 Marks
I.F. $= x$.
$yx = \int x \sin x \, dx + C = -x \cos x + \sin x + C$.
yx = sin x - x cos x + C
Q23
2010 Board
00:00
Solve the differential equation $\frac{dy}{dx} + \frac{y}{x} = x^2$. 3 Marks
I.F. $= x$.
$yx = x^4/4 + C$.
y = x³/4 + C/x
Q24
2009 Board
00:00
Solve the differential equation $x \frac{dy}{dx} = y + x^2 e^x$. 4 Marks
$\frac{dy}{dx} - \frac{1}{x}y = x e^x$.
I.F. $= 1/x$.
$y/x = e^x + C$.
y = x eˣ + Cx
Q25
2008 Board
00:00
Solve the differential equation $\frac{dy}{dx} = x + y$. 3 Marks
$\frac{dy}{dx} - y = x$.
I.F. $= e^{-x}$.
$y e^{-x} = -x e^{-x} - e^{-x} + C$.
y = -x - 1 + Ceˣ