Differential Equations - 5 - Mathematics MCQ

One of the points which lies on the solution curve of the differential equation (y – x) dx + (x + y) dy = 0, the curve passes through (0,1)

The solution of the differential equation  which vanishes when x = 0 and tends to a finite limit as  x →  -∞ is

If u₁, u₂, u₃ are solutions of the equation  then 2u₁ + au₂ + bu₂ and au₁ + bu₂ + u₃ are also solutions of given differential equation, then the values of a and b are 

The largest value of c such that there exists a function h(x) for -c < x < c that is a solution dy/dx = 1 +y² with h(0) = 0 is given by

Let y₁ and y₂ be two linearly independent solutions of y" + (sin x)y = 0, 0 ≤ x ≤ 1
Let g(x) = W(y₁, y₂) (x) be the Wronskian of y₁ and y₂.
Then,

If y₁(x) and y₂(x) are solutions of y" + x²y' - (1 - x) y = 0 such that y₁(0) = 0 ,y₁'(0 ) = -1 and y₂(0) = - 1, y₂' (0) = 1, then the Wronskian W(y₁, y₂) on R

For the sturm-Liouville problem (1 + x²)y" + 2xy' + λx²y = 0 with y'(1) = 0 and y'(10) = 0 the eigen values, λ satisfy

Consider the following statement for IVP  y(0) = 1 ,D : |x| ≤ 1, |y - 1 | ≤ 1
I. It has a solution which exists for all n
II. The local interval for which the solution exists uniquely is 
III. It has no solution in the interval |x| ≤ 3.
Then select the correct code.

Let y(x) be a non-trivial solution of the 2nd order linear differential equation

where C < 0; K > 0 and C² > K, then

Let y₁(x) and y₂(x) be two linearly independent solution of the differential equation  satisfying y₁(0) = 1, y₂(0) = -1 y'₁(0) = 1 and y'2( 0) = 1. Then, the Wronskian of y₁(x) and y₂(x) at x = 2, i.e.,  is equal to

The coordinate at a moving point P satisfying the equations   of the curve passes through the point (π/2, 0). When t = 0, then the equation of the curve in rectangular coordinates is

Let y₁ and y₂ be solutions of Bessel’s equation t²y" + ty' + (t² - n²)y = 0 on the interval 0 < t < ∝ with y₁(1) = 1, y'₁ (1) = 0, y₂(1) = 0 and y'₂(1) = 1, then value of W[y₁, y₂] (t) is

Differential equation |y'| + |y| = 0 has 

Let A be a 3 x 4 matrix of rank 2. Then the rank of ATA, where AT denotes the transpose of A. is,

The solution of y" + ay' + by = 0 where a and b are constants, approaches to zero as x → ∝ then

The primitive of the differential equation

For the initial value problem y' = f(x,y) with y(0)  = 0,
which of the following statement is true?