Tag : ordinary differential equation

For the differential equation y(8x - 9y)dx + 2x(x - 3y)dy = 0, which one of the following statements is TRUE?
Let y_c : R rightarrow (0, infty) be the solution of the Bernoulli’s equation dy/dx - y + y^3 = 0, y(0) = c > 0. Then, for every c > 0, which one of the following is true?
For 𝛼 ∈ ℝ, let 𝑦_𝛼(𝑥) be the solution of the differential equation dy/dx + 2y = 1/(1 + x^2) for x ∈ ℝ satisfying 𝑦(0) = 𝛼. Then, which one of the following is TRUE?
Let 𝑓: ℝ → ℝ be a solution of the differential equation d^2y/dx^2 - 2dy/dx + y = 2e^x for x ∈ ℝ. Consider the following statements. P: If 𝑓(𝑥) > 0 for all 𝑥 ∈ ℝ, then 𝑓′(𝑥) > 0 for all 𝑥 ∈ ℝ . Q: If 𝑓′(𝑥) > 0 for all 𝑥 ∈ ℝ, then 𝑓(𝑥) > 0 for all 𝑥 ∈ ℝ . Then, which one of the following holds?
Let y(x) be the solution of the differential equation dy/dx = 1 + y.sec{x}, for x in (-pi/2, pi/2) that satisfies y(0) = 0. Then, the value of y(pi/6) equals
Let 𝑦: ℝ → ℝ be the solution to the differential equation d^2y/dx^2 + 2dy/dx + 5y = 1 satisfying 𝑦(0) = 0 and 𝑦′(0) = 1. Then, lim_{x to infty}y(x) equals _____________ (rounded off to two decimal places).
For 𝛼 > 0, let 𝑦_𝛼(𝑥) be the solution to the differential equation 2d^2y/dx^2 - dy/dx - y = 0 satisfying the conditions 𝑦(0) = 1, 𝑦′(0) = 𝛼. Then, the smallest value of 𝛼 for which 𝑦_𝛼(𝑥) has no critical points in ℝ equals _____________ (rounded off to the nearest integer).
Let F be the family of curves given by x^2 + 2hxy + y^2 = 1, -1 < h < 1. Then, the differential equation for the family of orthogonal trajectories to F is
For 𝛼 ∈ (−2𝜋,0), consider the differential equation x^2·d^2y/dx^2 + 𝛼x·dy/dx + y = 0 for x > 0. Let 𝐷 be the set of all 𝛼 ∈ (−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 _____________
The solution of dy/dx = x/y is-
The degree of the equation ((d^2y)/(dx^2))^3-4(dy)/(dx)=2 is
The order of the differential equation dy/dx + 4y = 2x is-
The solution of the differential equation dy/dx = e^{x-y} is