List of top Differential Equations Questions on Differential Equations asked in IIT JAM MA

For some real number c with 0 < c < 1, let φ:(1-c, 1 + c) → (0,∞) be a differentiable function such that φ(1) = 1 and y = φ(x) is a solution of the differential equation (x² + y²)dx-4xy dy = 0. Then which one of the following is true?

Let S be the set of all continuous functions f: [-1,1]→\(\R\) satisfying the following three conditions:
(i) f is infinitely differentiable on the open interval (-1,1),
(ii) the Taylor series
\(f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+...\) of f at 0 converges to f(x) for each x ∈ (-1,1),
(iii) \(f(\frac{1}{n})=0\ \text{for all}\ n\isin\N\)
Then which of the following is/are true?

Consider the function u:R³→R given by
\(u(x_1,x_2,x_3)=x_1x^4_2x^2_3-x^3_1x^4_3-26x^2_1x^2_2x^3_3\)
Let CεN be such that
\(x_1\frac{ ∂u}{∂x_2}+2x_2\frac{ ∂u}{∂x_3} \)
evaluate at the point (t, t²,t³), eqals ct^k for every tεR, The n the value of K is equal to____.

Let y(x) be the solution of the deferential equation \(\frac{dy}{dx}+3x^2y=x^2\), for xεR,
satisfying the initial condition y(0)=4.
Then \(lim \,\,y_{n→∞}\) y(x) is equal to ____ (Rounded off to decimal places)

Let y(x) be the solution of the differential equation
\(x,y^2y'+y^3=\frac{sin x}{x} for x>0\)
satisfying (\(\frac{π}{2}\))=0
The the value of y (\(\frac{5π}{2}\)) is equal to ____. (Rounded off to two decimal places)

Consider the region G={(x,y,z) εR³: 0<z<x²-y², x²+y²<1}.
Then the volume of G is equal to____. (Rounded off to two decimal place).

Consider the family F1 of curve lying in the region
{(x,y) εR²: y>0 and 0<x<π }
and given by
\(y=\frac{c(1-cosx)}{sinx},\) Where c is a positive real number.
Let F₂ be the family of orthogonal trajectories to f₁. consider the curve C belonging to the family F2 passing through the point (\(\frac{π}{3}\),1). if a is a real number such that (\(\frac{π}{4}\), a) lies on C, then the value of a⁴ is equal _____to (Rounded off to two decimal places).

Given that y(x) is a solution of the deferential equation
\(x^2y''+xy'-4y-x^2\)
on the interval (0,∞) such that the \(lim_{x→0^+}\) y(x) exists and y(1)=1. the value y'(1) is qual to____.(Rounded off to two decimal places).

Let f : (−1, 1) → \(\R\) and g : (−1, 1) → \(\R\) be thrice continuously differentiable functions such that f(x) ≠ g(x) for every nonzero x ∈ (−1, 1). Suppose
f(0) = ln 2, f'(0) = π, f"(0) = π² , and f"'(0) = π⁹
and
g(0) = ln 2, g'(0) = π, g′′(0) = π², and g'"(0) = π³.
Then the value of the limit
\(\lim\limits_{x\rightarrow0}\frac{e^{f(x)}-e^{g(x)}}{f(x)-g(x)}\)
is equal to _________. (Rounded off to two decimal places)

Let (-c,c) be the largest open interval in \(\R\) (where c is either a positive real number or c = ∞) on which the solution y(x) of the differential equation \(\frac{dy}{dx}=x^2+y^2+1\) with initial condition y(0) = 0 exists and is unique. Then which of the following is/are true?

Let S be the set of all functions f: \(\R\rightarrow\R\) satisfying \(|f(x)-f(y) |^2 \le |x - y|^3\ for\ all\ x, y \isin \R.\) Then which of the following is/are true?

If \( x^2 + xy^2 = c \), where \( c \in \mathbb{R} \), is the general solution of the exact differential equation \[ M(x, y)\, dx + 2xy\, dy = 0, \] then \( M(1,1) \) is ............

A solution of the differential equation \( 2x^2\dfrac{d^2y}{dx^2} + 3x\dfrac{dy}{dx} - y = 0, \, x > 0 \) that passes through the point (1, 1) is

Consider the differential equation \( L[y] = (y - y^2)dx + xdy = 0. \) The function \( f(x, y) \) is said to be an integrating factor of the equation if \( f(x, y)L[y] = 0 \) becomes exact. If \( f(x, y) = \dfrac{1}{x^2 y^2}, \) then

If \( u = x^3 \) and \( v = y^2 \) transform the differential equation \( 3x^5 dx - y(y^2 - x^3) dy = 0 \) to \( \dfrac{dv}{du} = \dfrac{\alpha u}{2(u - v)} \), then \( \alpha \) is

Let \( f : \mathbb{R} \to \mathbb{R} \) be a differentiable function with \( f'(x) = f(x) \) for all \( x. \) Suppose that \( f(\alpha x) \) and \( f(\beta x) \) are two non-zero solutions of the differential equation \[ 4 \frac{d^2 y}{dx^2} - p \frac{dy}{dx} + 3y = 0 \] satisfying \( f(\alpha x)f(\beta x) = f(2x) \) and \( f(\alpha x)f(-\beta x) = f(x). \) Then, the value of \( p \) is ...........

Let \( f(x, y) = 0 \) be a solution of the homogeneous differential equation \( (2x + 5y)dx - (x + 3y)dy = 0. \)
If \( f(x + \alpha, y - 3) = 0 \) is a solution of \( (2x + 5y - 1)dx + (2 - x - 3y)dy = 0, \) then the value of \( \alpha \) is .............

Let \( L[y] = x^2\dfrac{d^2y}{dx^2} + px\dfrac{dy}{dx} + qy, \) where \( p, q \) are real constants. Let \( y_1(x) \) and \( y_2(x) \) be two solutions of \( L[y] = 0, \, x > 0, \) that satisfy \( y_1(x_0) = 1, y_1'(x_0) = 0, y_2(x_0) = 0, y_2'(x_0) = 1 \) for some \( x_0 > 0. \) Then,

Consider the differential equation \[ \frac{dy}{dx} + 10y = f(x), x > 0, \] where \( f(x) \) is a continuous function such that \( \lim_{x \to \infty} f(x) = 1. \) Then the value of \( \lim_{x \to \infty} y(x) \) is .................

The solution(s) of the differential equation \( \frac{dy}{dx} = (\sin 2x) y^{1/3}\) satisfying \(y(0) = 0\) is(are)

A particular integral of the differential equation
\[ y'' + 3y' + 2y = e^{e^x} \] is

An integrating factor of the differential equation
\[ \left( y + \frac{1}{3} y^3 + \frac{1}{2} x^2 \right) \, dx + \frac{1}{4} (x + xy^2) \, dy = 0 \] is

Let \( y(x) \) be the solution of the differential equation \( \frac{dy}{dx} + y = f(x) \), for \( x \geq 0, y(0) = 0 \), where
\[ f(x) = \begin{cases} 2, & 0 \leq x < 1 \\ 0, & x \geq 1 \end{cases} \] Then \( y(x) \) is

In \( \mathbb{R}^2 \), the family of trajectories orthogonal to the family of asteroids \( x^{2/3} + y^{2/3} = a^{2/3} \) is given by

Let \( f : [0, \infty) \to [0, \infty) \) be continuous on \( [0, \infty) \) and differentiable on \( (0, \infty) \). If \[ f(x) = \int_0^x \sqrt{f(t)} \, dt, \text{ then } f(6) = \, \text{..........} \]