Question:

Let 
S=(0,2π){π2,3π4,3π2,7π4}S = (0, 2\pi) - \left\{\frac{\pi}{2}, \frac{3\pi}{4}, \frac{3\pi}{2}, \frac{7\pi}{4}\right\}
. Let y = y(x), x∈S, be the solution curve of the differential equation 
dydx=11+sin2x,y(π4)=12\frac{dy}{dx}=\frac{1}{1+sin⁡2x},y(\frac{π}{4})=\frac{1}{2}
 .If the sum of abscissas of all the points of intersection of the curve y = y(x) with the curve 
y=2sinxy=\sqrt2sin⁡x is kπ12\frac{kπ}{12},
 then k is equal to _________.

Updated On: Mar 2, 2024
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Correct Answer: 42

Solution and Explanation

The correct answer is 42
dydx=11+sin2x\frac{dy}{dx}=\frac{1}{1+sin⁡2x}
dy=sec2xdx(1+tanx)2⇒dy=\frac{sec^2⁡xdx}{(1+tan⁡ x)^2}
y=11+tanx+c⇒y=−\frac{1}{1+tan⁡x}+c
When
x=π4,y=12x=\frac{π}{4}, y=\frac{1}{2} gives c=1
So y=tanx1+tanxy=sinxsinx+cosxy=\frac{tan⁡x}{1+tan⁡x}⇒y=\frac{sin⁡x}{sin⁡x+cos⁡x}
Now, y=2sinxsinx=0y=\sqrt2sin⁡x ⇒sin⁡x=0
or
sinx+cosx=12sin⁡x+cos⁡x=\frac{1}{\sqrt2}
sinx = 0 gives x = π only
and
sinx+cosx=12sin(x+π4)=12sin⁡x+cos⁡x=\frac{1}{\sqrt2}⇒sin⁡(x+\frac{π}{4})=\frac{1}{2}
So
x+π4=5π6x+\frac{π}{4}=\frac{5π}{6} or 13π6x=7π12\frac{13π}{6}⇒x=\frac{7π}{12} or 23π12\frac{23π}{12}

Sum of all solutions = π+7π12+23π12=42π12= π+\frac{7π}{12}+\frac{23π}{12}=\frac{42π}{12}
Therefore,  k = 42.

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Concepts Used:

Differential Equations

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.

Orders of a Differential Equation

First Order Differential Equation

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’

Second-Order Differential Equation

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”.

Types of Differential Equations

Differential equations can be divided into several types namely

  • Ordinary Differential Equations
  • Partial Differential Equations
  • Linear Differential Equations
  • Nonlinear differential equations
  • Homogeneous Differential Equations
  • Nonhomogeneous Differential Equations