Let the solution curve y = y(x) of the differential equation (4 + x2)dy – 2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.
Let the solution curve y = y(x) of the differential equation (4 + x2)dy – 2x(x2 + 3y + 4)dx = 0 pass through the origin. Then y(2) is equal to _______.
Correct Answer: 12
Solution and Explanation
The correct answer is 12
(4 + x2) dy – 2x(x2 + 3y + 4)dx = 0
\(⇒\frac{dy}{dx}=(\frac{6x}{x^2+4})y+2x\)
\(⇒\frac{dy}{dx}−(\frac{6x}{x^2+4})y=2x\)
I.F. \(= e^{−3 In(x^2+4)}=\frac{1}{(x^2+4)^3}\)
\(So, \frac{y}{(x^2+4)^3}=∫\frac{2x}{(x^2+4)^3}dx+c\)
\(⇒y=−\frac{1}{2}(x^2+4)+c(x^2+4)^3\)
When x = 0, y = 0 gives
\(c=\frac{1}{32}\)
So, for x = 2,y = 12
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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