Differential Equations — JEE Main PYQs

243 previous year questions on Differential Equations (Mathematics), newest first. Open any question for its complete step-by-step solution.

Let $y = y (x)$ be the solution of the differential equation $(1 + x^{2}) \frac{d y}{d x} + y = e^{t a n^{- 1} x}$ , $y (1) = 0$ . Then $y (0)$ is
JEE Main 2024View solution →
Suppose the solution of the differential equation $\frac{d y}{d x} = \frac{( 2 + α ) x - β y + 2}{β x - 2 α y - ( β γ - 4 α )}$ represents a circle passing through origin. Then radius of the circle is
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If the solution $y (x)$ of the given differential equation $(e^{y} + 1) cos x d x + e^{y} sin x d y = 0$ passes through the point $(\frac{π}{2}, 0)$ then the value of $e^{y (\frac{π}{6})}$ is equal to ______.
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Let $f (x)$ be a positive function such that the area bounded by $y = f (x), y = 0$ from $x = 0$ to $x = a > 0$ is $e^{- a} + 4 a^{2} + a - 1$ . Then the differential equation, whose general solution is $y = c_{1} f (x) + c_{2}$ , …
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Let $y = y (x)$ be the solution of the differential equation $(1 + y^{2}) e^{t a n x} d x + cos^{2} x (1 + e^{2 t a n x}) d y = 0$ , $y (0) = 1$ . Then $y (\frac{π}{4})$ is equal to :
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Let $y = y (x)$ be the solution curve of the differential equation $sec y \frac{d y}{d x} + 2 x sin y = x^{3} cos y$ , $y (1) = 0$ . Then $y (3)$ is equal to :
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Let $α ∣ x ∣ = ∣ y ∣ e^{x y - β}$ , $α, β \in N$ be the solution of the differential equation $x d y - y d x + x y (x d y + y d x) = 0$ , $y (1) = 2$ . Then $α + β$ is equal to ______.
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The solution curve, of the differential equation $2 y \frac{dy}{dx} + 3 = 5 \frac{dy}{dx}$ , passing through the point $(0, 1)$ is a conic, whose vertex lies on the line :
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The solution of the differential equation
$(x^{2} + y^{2}) d x - 5 x y d y = 0, y (1) = 0$ , is :
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Let $\int_{0}^{x} 1 - (y^{'} (t))^{2} dt = \int_{0}^{x} y (t) dt, 0 \leq x \leq 3, y \geq 0$ , $y (0) = 0$ . Then at $x = 2, y^{''} + y + 1$ is equal to :
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For a differentiable function $f : I R \to I R$ , suppose $f^{'} (x) = 3 f (x) + α$ , where $α \in IR, f (0) = 1$ and $lim_{x \to - \infty} f (x) = 7$ . Then $9 f (- lo g_{e} 3)$ is equal to
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If $sin (\frac{y}{x}) = lo g_{e} ∣ x ∣ + \frac{α}{2}$ is the solution of the differential equation $x cos (\frac{y}{x}) \frac{d y}{d x} = y cos (\frac{y}{x}) + x$ and $y (1) = \frac{π}{3}$ , then $α^{2}$ is equal to
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Let $f (x) = lim_{r \to x} {\frac{2 r ^{2} [( f ( r ) ) ^{2} - f ( x ) f ( r )]}{r ^{2} - x ^{2}} - r^{3} e^{\frac{f ( r )}{r}}}$ be differentiable in $(- \infty, 0) \cup (0, \infty)$ and $f (1) = 1$ . Then the value of $e a$ , such that $f (a) = 0$ , is equal to ______.
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The temperature $T (t)$ of a body at time $t = 0$ is $16 0^{\circ}$ F and it decreases continuously as per the differential equation $\frac{dT}{dt} = - K (T - 80)$ , where K is positive constant. If $T (15) = 12 0^{\circ}$ …
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Let $y = y (x)$ be the solution of the differential equation
$sec^{2} x d x + (e^{2 y} tan^{2} x + tan x) d y = 0,$

$0 < x < \frac{π}{2}$ , $y (\frac{π}{4}) = 0$ . If $y (\frac{π}{6}) = α$ ,

Then $e^{8 α}$ is equal to ______ …
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Let $y = y (x)$ be the solution of the differential equation $sec x d y + {2 (1 - x) tan x + x (2 - x)} d x = 0$ such that $y (0) = 2$ . Then $y (2)$ is equal to :
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Let $y = y (x)$ be the solution of the differential equation $(1 - x^{2}) d y = [x y + (x^{3} + 2) 3 (1 - x^{2})] d x$ , $- 1 < x < 1$ , $y (0) = 0$ . If $y (\frac{1}{2}) = \frac{m}{n}$ , $m$ and $n$ are coprime numbers, then $m + n$ is equal to ______.
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If $y = y (x)$ is the solution curve of the differential equation $(x^{2} - 4) d y - (y^{2} - 3 y) d x = 0$ , $x > 2, y (4) = \frac{3}{2}$ and the slope of the curve is never zero, then the value of $y (10)$ equals :
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If the solution curve, of the differential equation $\frac{d y}{d x} = \frac{x + y - 2}{x - y}$ passing through the point $(2, 1)$ is $tan^{- 1} (\frac{y - 1}{x - 1}) - \frac{1}{β} lo g_{e} (α + (\frac{y - 1}{x - 1})^{2}) = lo g_{e} ∣ x - 1∣$ , then …
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Let $x = x (t)$ and $y = y (t)$ be solutions of the differential equations $\frac{d x}{d t} + a x = 0$ and $\frac{d y}{d t} + b y = 0$ respectively, $a, b \in R$ . Given that $x (0) = 2$ ; $y (0) = 1$ and $3 y (1) = 2 x (1)$ , the value of $t$ , for which $x (t) = y (t)$ , is :
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If the solution of the differential equation $(2 x + 3 y - 2) d x + (4 x + 6 y - 7) d y = 0, y (0) = 3$ , is $α x + β y + 3 lo g_{e} ∣2 x + 3 y - γ ∣ = 6$ , then $α + 2 β + 3 γ$ is equal to ______.
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If $\frac{dx}{dy} = \frac{1 + x - y ^{2}}{y}, x (1) = 1$ , then $5 x (2)$ is equal to :
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The solution curve of the differential equation
$y \frac{d x}{d y} = x (lo g_{e} x - lo g_{e} y + 1), x > 0, y > 0$ passing
through the point ( $e, 1$ ) is
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Let $y = y (x)$ be the solution of the differential equation
$\frac{dy}{dx} = \frac{( tan x ) + y}{sin x ( sec x - sin x tan x )}$

$x \in (0, \frac{π}{2})$ satisfying the condition …
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Let $y = y (x)$ be the solution of the differential equation $\frac{dy}{dx} = 2 x (x + y)^{3} - x (x + y) - 1, y (0) = 1$ . Then, $(\frac{1}{2} + y (\frac{1}{2}))^{2}$ eq …
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If $x = x (t)$ is the solution of the differential equation $(t + 1) dx = (2 x + (t + 1)^{4}) dt, x (0) = 2$ , then, $x (1)$ equals ____.
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A function $y = f (x)$ satisfies $f (x) sin 2 x + sin x - (1 + cos^{2} x) f^{'} (x) = 0$ with condition $f (0) = 0$ . Then $f (\frac{π}{2})$ is equal to
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If the solution curve $y = y (x)$ of the differential equation $(1 + y^{2}) (1 + lo g_{e} x) d x + x d y = 0, x > 0$ passes through the point $(1, 1)$ and $y (e) = \frac{α - t a n ( \frac{3}{2} )}{β + t a n ( \frac{3}{2} )}$ , then $α + 2 β$ is
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Let $Y = Y (X)$ be a curve lying in the first quadrant such that the area enclosed by the line $Y - y = Y^{'} (x) (X - x)$ and the co-ordinate axes, where ( $x, y$ ) is any point on the curve, is always …
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If the solution curve $f (x, y) = 0$ of the differential equation $(1 + lo g_{e} x) \frac{d x}{d y} - x lo g_{e} x = e^{y}$ , $x > 0$ , passes through the points $(1, 0)$ and $(α, 2)$ then $α^{α}$ is equal to
JEE Main 2023View solution →
Let a curve $y = f (x), x \in (0, \infty)$ pass through the points $P (1, \frac{3}{2})$ and $Q (a, \frac{1}{2})$ . If the tangent at any point $R (b, f (b))$ to the given curve cuts the y-axis at the point $S (0, c)$ such that $b c = 3$ , then $(P Q)^{2}$ is equal to ____.
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If the solution curve of the differential equation $(y - 2 lo g_{e} x) d x + (x lo g_{e} x^{2}) d y = 0, x > 1$ passes through the points $(e, \frac{4}{3})$ and $(e^{4}, α)$ , then $α$ is equal to ______.
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Let $y = y (x)$ be a solution of the differential equation $(x cos x) d y + (x y sin x + y cos x - 1) d x = 0$ , $0 < x < \frac{π}{2}$ . If $\frac{π}{3} y (\frac{π}{3}) = 3$ , then …
JEE Main 2023View solution →
Let the solution curve $x = x (y)$ , $0 < y < \frac{π}{2}$ , of the differential equation $(lo g_{e} (cos y))^{2} cos y d x - (1 + 3 x lo g_{e} (cos y)) sin y d y = 0$ satisfy $x (\frac{π}{3}) = \frac{1}{2 l o g _{e} 2}$ . If $x (\frac{π}{6}) = \frac{1}{l o g _{e} m - l o g _{e} n}$ , w …
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Let the tangent at any point P on a curve passing through the points $(1, 1)$ and $(\frac{1}{10}, 100)$ , intersect positive x-axis and y-axis at the points A and B respectively. If $PA : PB = 1 : k$ and $y = y (x)$ …
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Let $f$ be a differentiable function such that $x^{2} f (x) - x = 4 \int_{0}^{x} t f (t) d t, f (1) = \frac{2}{3}$ .

Then $18 f (3)$ is equal to
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The slope of tangent at any point $(x, y)$ on a curve $y = y (x)$ is $\frac{x ^{2} + y ^{2}}{2 x y}, x > 0$ . If $y (2) = 0$ , then a value of $y (8)$ is
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Let $y = y (x), y > 0$ , be a solution curve of the differential equation $(1 + x^{2}) d y = y (x - y) d x$ . If $y (0) = 1$ and $y (22) = β$ , then
JEE Main 2023View solution →
If $y = y (x)$ is the solution of the differential equation $\frac{d y}{d x} + \frac{4 x}{( x ^{2} - 1 )} y = \frac{x + 2}{( x ^{2} - 1 ) ^{\frac{5}{2}}}, x > 1$ such that $y (2) = \frac{2}{9} lo g_{e} (2 + 3)$ and …
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Let $y = y_{1} (x)$ and $y = y_{2} (x)$ be the solution curves of the differential equation $\frac{d y}{d x} = y + 7$ with initial conditions $y_{1} (0) = 0$ , $y_{2} (0) = 1$ respectively. Then the curves $y = y_{1} (x)$ and $y = y_{2} (x)$ intersect at
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Let $x = x (y)$ be the solution of the differential equation $2 (y + 2) lo g_{e} (y + 2) dx + (x + 4 - 2 lo g_{e} (y + 2)) dy = 0$ , $y > - 1$ with $x (e^{4} - 2) = 1$ . Then …
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Let $y = y (x)$ be the solution of the differential equations $\frac{d y}{d x} + \frac{5}{x ( x ^{5} + 1 )} y = \frac{( x ^{5} + 1 ) ^{2}}{x ^{7}}, x > 0$ . If $y (1) = 2$ , then $y (2)$ is equal to
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Let $y = y (x)$ be a solution curve of the differential equation, $(1 - x^{2} y^{2}) d x = y d x + x d y$ .

If the line $x = 1$ intersects the curve $y = y (x)$ at $y = 2$ and the line $x = 2$ intersects the curve …
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Let $y = y (x)$ be the solution of the differential equation $x^{3} d y + (x y - 1) d x = 0, x > 0$ , $y (\frac{1}{2}) = 3 - e$ . Then $y (1)$ is equal to
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Let the solution curve $y = y (x)$ of the differential equation
$\frac{d y}{d x} - \frac{3 x ^{5} tan ^{- 1} ( x ^{3} )}{( 1 + x ^{6} ) ^{\frac{3}{2}}} y = 2 x exp \frac{x ^{3} - tan ^{- 1} x ^{3}}{1 + x ^{6}}$

pass through the origin. Then $y (1)$ is equal to:
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Let $α x = exp (x^{β} y^{γ})$ be the solution of the differential equation $2 x^{2} y d y - (1 - x y^{2}) d x = 0$ , $x > 0, y (2) = lo g_{e} 2$ . Then $α + β - γ$ equals :
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Let $y = y (x)$ be the solution of the differential equation $x lo g_{e} x \frac{d y}{d x} + y = x^{2} lo g_{e} x$ , $(x > 1)$ . If $y (2) = 2$ , then $y (e)$ is equal to
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Let $y = f (x)$ be the solution of the differential equation $y (x + 1) d x - x^{2} d y = 0, y (1) = e$ . Then $lim_{x \to 0^{+}} f (x)$ is equal to
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Let $y = y (t)$ be a solution of the differential equation
$\frac{d y}{d t} + α y = γ e^{- β t}$
Where, $α > 0, β > 0$ and $γ > 0$ . Then $Lim_{t \to \infty} y (t)$
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Let $y = y (x)$ be the solution of the differential equation $(x^{2} - 3 y^{2}) d x + 3 x y d y = 0, y (1) = 1$ . Then $6 y^{2} (e)$ is equal to
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