List of top Differential Equations Questions

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?

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).

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).

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).

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 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?

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?