Differential Equations - 4 - Mathematics MCQ

The initial value problem

Consider the following statements
I. A singular solution of differential equation satisfies the differential equation but is not a particular solution of the equation.
II. If T(x, y) = 0 is the equation of the tac-locus, then
T(x, y ) is a factor of the P-discriminant.
III. T(x, y ) is a factor of a c-discriminant.
IV. Cusp-locus has two distinct tangent.
Choose the correct answer.

The general solution 
where  andy(0) = 0 is

The solution of the differential equation 

Singular solution of differential equation contains
I. arbitrary constant
II. can be obtained from general
III. do not contain arbitrary constant
IV. cannot be obtained from general solution

For non-homogeneous equation y' + p(x)y = r(x), if y₁ and y₂ are its solutions, then the solution of homogeneous equation y' + p(x)y = 0 is

The solution of  represents

Consider the differential equation (x + y + 1) dx + (2x + 2y + 1)dy = 0. Which of the following statement is true? 

The orthogonal trajectory of the family x² - y² = C₂ are given by

A. Singular solution contains no arbitrary constants.
B. Singular solution can be obtained from complete primitive.

Let the general solution of a differential equation be, y = ae^bx+c then order of the differential equation is,

If y₁(x) = x and y₂(x) = xe^x are two linearly independent solutions of  then the interval on which they form a fundam ental set of solution is

Using the method of variation of parameters for the particular solution to the differential equation

The general solution of the system of differential equation  and M a 2 x 2 matrix  and a 2 x 1 constant vector b =  is given by

For the ordinary differential equation

which of the following statement is true?

If 2x(1 - y ) = K and g(x, y) = L are orthogonal families of curves where K and L constants, then g(x, y) is

If y₁ and y₂ are linearly independent solutions of the homogeneous equation L(y) =y"+p₁(x)y' + p₂(x)y = 0. Then, p₁(x) and p₂{x) are given by