List of top Differential Equations Questions

Let L₁ denote the line y = 3x + 2 and L₂ denote the line y = 4x + 3. Suppose that f: ℝ → ℝ is a four times continuously differentiable function such that the line L₁ intersects the curve y = f(x) at exactly three distinct points and the line L₂ intersects the curve y = f(x) at exactly four distinct points. Then, which one of the following is TRUE ?

For a ∈ ℝ, let ya(x) be the solution of the differential equation
\(\frac{dy}{dx}+2y=\frac{1}{1+x^2}\) for x ∈ \(\R\)
satisfying y(0) = a. Then, which one of the following is TRUE ?

Let y : ℝ → ℝ be the solution to the differential equation
\(\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=1\)
satisfying y(0) = 0 and y'(0) = 1.
Then, \(\lim\limits_{x \rightarrow \infin}y(x)\) equals __________ (rounded off to two decimal places).

For a ∈ (−2π, 0), consider the differential equation
\(x^2\frac{d^2y}{dx^2}+ax\frac{dy}{dx}+y=0\) for x > 0.
Let D be the set of all a ∈ (−2π, 0) for which all corresponding real solutions to the above differential equation approach zero as x → 0⁺. Then, the number of elements in D ∩ \(\Z\) equals ___________

Let
S = {(x, y, x) ∈ \(\R^3\) ∶ x² + y² + z² = 4, (x - 1)² + y² ≤ 1, 𝑧 ≥ 0}.
Then, the surface area of S equals ___________ (rounded off to two decimal places).

For a > 0, let y_a(x) be the solution to the differential equation
\(2\frac{d^2y}{dx^2}-\frac{dy}{dx}-y=0\)
satisfying the conditions
y(0) = 1, y'(0) = a.
Then, the smallest value of a for which ya(x) has no critical points in ℝ equals ___________ (rounded off to the nearest integer).

For the differential equation
y(8x − 9y)dx + 2x(x − 3y)dy = 0 ,
which one of the following statements is TRUE ?

Let \(y : (\sqrt{\frac{2}{3}}, ∞) → \R\) be the solution of
(2x − y)y' + (2y − x) = 0,
y(1) = 3.
Then

For a twice continuously differentiable function 𝑔: ℝ → ℝ, define
\(u_g(x,y)=\frac{1}{y}\int^y_{-y}g(x+t)dt\ \ \ \text{for}(x,y)\in \R^2, \ \ \ y \gt0.\)
Which one of the following holds for all such 𝑔 ?

Let \(y_c:\R \rightarrow(0,\infin)\) be the solution of the Bernoulli’s equation
\(\frac{dy}{dx}-y+y^3=0,\ \ \ \ \ \ \ y(0)=c \gt 0.\)
Then, for every 𝑐 > 0, which one of the following is true ?

Let f(x, y) = ln(1 + x² + y² ) for (x, y) ∈ \(\R^2\). Define
\(\begin{matrix} P=\frac{∂^2f}{∂x^2}|_{(0,0)} &  Q=\frac{∂^2f}{∂x∂y}|_{(0,0)} \\  R=\frac{∂^2f}{∂y∂x}|_{(0,0)} &  S=\frac{∂^2f}{∂y^2}|_{(0,0)} \end{matrix}\)
Then

Let y : \(\R → \R\) be a twice differentiable function such that y" is continuous on [0, 1] and y(0) = y(1) = 0. Suppose y"(x) + x² < 0 for all x ∈ [0, 1]. Then

Let y : (1, ∞) → \(\R\) be the solution of the differential equation
\(y"-\frac{2y}{(1-x)^2}=0\)
satisfying y(2) = 1 and\(\lim\limits_{x→∞}y(x) = 0\). Then y(3) is equal to __________. (rounded off to two decimal places)

If y is the solution of
y" - 2y' + y = e^x,
y(0) = 0, y'(0) = −1/2,
then y(1) is equal to __________. (rounded off to two decimal places)

Consider the family of curves x² + y² = 2x + 4y + k with a real parameter k > −5. Then the orthogonal trajectory to this family of curves passing through (2, 3) also passes through

Consider the initial value problem
\(\frac{dy}{dx}+αy=0, \\ y(0)=1,\)
where α ∈ \(\R\). Then

Let f : \(\R^3 → \R\) be defined as f(x, y, z) = x³ + y³ + z³ , and let L : \(\R^3 → \R\) be the linear map satisfying
\(\lim\limits_{(x,y,z)\rightarrow(0,0,0)}\frac{f(1+x,1+y,1+z)-f(1,1,1)-L(x,y,z)}{\sqrt{x^2+y^2+z^2}}=0.\)
Then L(1, 2, 4) is equal to ____________. (rounded off to two decimal places)

Let 𝑦(𝑥) be the solution of the differential equation
\(\frac{dy}{dx}=1+y\sec x\ \ \text{for}\ x \in(-\frac{\pi}{2},\frac{\pi}{2})\)
that satisfies 𝑦(0) = 0. Then, the value of \(y(\frac{\pi}{6})\) equals

Let 𝑔: ℝ → ℝ be a continuous function. Which one of the following is the solution of the differential equation
\(\frac{d^2y}{dx^2}+y=g(x)\ \ \ \ \text{for}\ x \in \R\),
satisfying the conditions y(0) = 0, y'(0) = 1 ?

Let F be the family of curves given by
x² + 2hxy + y² = 1, -1 < h < 1.
Then, the differential equation for the family of orthogonal trajectories to F is

On the open interval \((-c, c)\), where \( c \) is a positive real number, \( y(x) \) is an infinitely differentiable solution of the differential equation\[ \frac{dy}{dx} = y^2 - 1 + \cos x, \] with the initial condition \( y(0) = 0 \). Then which one of the following is correct?

Let \(\theta \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right)\). Consider the functions
\[ u : \mathbb{R}^2 - \{ (0, 0) \} \to \mathbb{R} \quad \text{and} \quad v : \mathbb{R}^2 - \{ (0, 0) \} \to \mathbb{R} \]
given by
\[ u(x,y) = x - \frac{x}{x^2 + y^2} \quad \text{and} \quad v(x,y) = y + \frac{y}{x^2 + y^2}. \]
The value of the determinant \(\begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}\) at the point \((\cos \theta, \sin \theta)\) is equal to

Consider the open rectangle \( G = \{ (s,t) \in \mathbb{R}^2 : 0 < s < 1 \ \text{and} \ 0 < t < 1 \} \) and the map \( T : G \to \mathbb{R}^2 \) given by \[ T(s,t) = \left( \frac{\pi s (1-t)}{2}, \ \frac{\pi (1-s)t}{2} \right) \ \text{for} \ (s,t) \in G. \] Then the area of the image \( T(G) \) of the map \( T \) is equal to

Let \( u: \mathbb{R} \to \mathbb{R} \) be a twice continuously differentiable function such that \( u(0) > 0 \) and \( u'(0) > 0 \). Suppose \( u \) satisfies \[ u''(x) = \frac{u(x)}{1 + x^2} \]
for all \( x \in \mathbb{R} \).
Consider the following two statements:
I. The function \( uu' \) is monotonically increasing on \([0, \infty)\).
II. The function \( u \) is monotonically increasing on \([0, \infty)\).
Then which one of the following is correct?

consider the differential equation
y''+ay'+y=sin x for x for xεR. (**) 
then which one of the following is true?