Variable Separable DEs PYQs
Practice Class 12 CBSE Board Previous Year Questions (2008-2026)
Q1
2008 Board
00:00
Solve the differential equation $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$. 2 Marks
Step 1: Separate Variables
$\frac{dy}{1+y^2} = \frac{dx}{1+x^2}$
Step 2: Integrate Both Sides
$\int \frac{dy}{1+y^2} = \int \frac{dx}{1+x^2}$
Step 3: Solve
$\tan^{-1} y = \tan^{-1} x + C$.
tan⁻¹ y = tan⁻¹ x + C
Q2
2013 Board
00:00
Solve the differential equation $\frac{dy}{dx} = e^{x-y}$. 2 Marks
Step 1: Separate Variables
$\frac{dy}{dx} = e^x \cdot e^{-y} \Rightarrow e^y \, dy = e^x \, dx$.
Step 2: Integrate
$\int e^y \, dy = \int e^x \, dx$
Step 3: Final Answer
$e^y = e^x + C$.
e^y = e^x + C
Q3
2017 Board
00:00
Solve the differential equation $\frac{dy}{dx} = 1 + x + y + xy$. 4 Marks
Step 1: Factorize RHS
$\frac{dy}{dx} = (1+x) + y(1+x) = (1+x)(1+y)$.
Step 2: Separate Variables
$\frac{dy}{1+y} = (1+x) \, dx$.
Step 3: Integrate
$\int \frac{dy}{1+y} = \int (1+x) \, dx$
$\log|1+y| = x + \frac{x^2}{2} + C$.
log|1+y| = x + x²/2 + C
Q4
2020 Board
00:00
Solve the differential equation $\frac{dy}{dx} = xy$. 2 Marks
Step 1: Separate Variables
$\frac{dy}{y} = x \, dx$.
Step 2: Integrate
$\int \frac{dy}{y} = \int x \, dx$
$\log|y| = \frac{x^2}{2} + C$.
log|y| = x²/2 + C
Q5
2019 Board
00:00
Solve the differential equation $\frac{dy}{dx} = e^{x+y}$. 2 Marks
Step 1: Separate Variables
$\frac{dy}{dx} = e^x \cdot e^y \Rightarrow e^{-y} \, dy = e^x \, dx$.
Step 2: Integrate
$-e^{-y} = e^x + C \Rightarrow e^x + e^{-y} = C'$.
eˣ + e⁻ʸ = C
Q6
2016 Board
00:00
Solve the differential equation $\frac{dy}{dx} = x + y + 1$. 4 Marks
Step 1: Substitution
Let $x+y+1 = v$. Then $1 + \frac{dy}{dx} = \frac{dv}{dx} \Rightarrow \frac{dy}{dx} = \frac{dv}{dx} - 1$.
Step 2: Transform
$\frac{dv}{dx} - 1 = v \Rightarrow \frac{dv}{dx} = v + 1$.
Step 3: Separate and Integrate
$\frac{dv}{v+1} = dx \Rightarrow \log|v+1| = x + C$.
Step 4: Substitute Back
$\log|x+y+2| = x + C$.
log|x+y+2| = x + C
Q7
2014 Board
00:00
Solve the differential equation $\frac{dy}{dx} = \frac{x}{y}$. 1 Mark
Step 1: Separate Variables
$y \, dy = x \, dx$.
Step 2: Integrate
$\frac{y^2}{2} = \frac{x^2}{2} + C \Rightarrow y^2 - x^2 = C'$.
y² - x² = C
Q8
2011 Board
00:00
Solve the differential equation $\frac{dy}{dx} = \frac{x+y+4}{x+y+2}$. 4 Marks
Step 1: Substitution
Let $x+y = v \Rightarrow 1 + \frac{dy}{dx} = \frac{dv}{dx} \Rightarrow \frac{dy}{dx} = \frac{dv}{dx} - 1$.
Step 2: Transform
$\frac{dv}{dx} - 1 = \frac{v+4}{v+2} \Rightarrow \frac{dv}{dx} = \frac{v+4+v+2}{v+2} = \frac{2v+6}{v+2}$.
Step 3: Separate and Integrate
$\frac{v+2}{2(v+3)} \, dv = dx \Rightarrow \frac{1}{2} \int (1 - \frac{1}{v+3}) \, dv = \int dx$.
$\frac{1}{2} (v - \log|v+3|) = x + C$.
Substitute $v=x+y$: $x+y - \log|x+y+3| = 2x + C' \Rightarrow y-x - \log|x+y+3| = C'$.
y - x - log|x+y+3| = C
Q9
2025 Board
00:00
Solve the differential equation $\frac{dy}{dx} = e^{x-y}$. 2 Marks
$e^y \, dy = e^x \, dx \Rightarrow e^y = e^x + C$.
e^y = e^x + C