Let
. Let y = y(x), x∈S, be the solution curve of the differential equation
.If the sum of abscissas of all the points of intersection of the curve y = y(x) with the curve
is ,
then k is equal to _________.
The correct answer is 42
When
gives c=1
So
Now,
or
sinx = 0 gives x = π only
and
So
or or
Sum of all solutions
Therefore, k = 42.
Let be a twice differentiable function such that for all . If and satisfies , where , then the area of the region R = {(x, y) | 0 y f(ax), 0 x 2\ is :
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Differential equations can be divided into several types namely