General solution of $y \frac{dy}{dx}+by^{2}=a\,cos\,x, 0 < x< 1$ is
- $y^{2} = 2a\left(2b\, sinx + cosx\right) + ce^{-2bx}$
- $\left(4b^{2} + 1\right)y^{2} = 2a\left(sinx + 2bcosx\right) + ce^{-2bx}$
- $\left(4b^{2} + 1\right)y^{2} = 2a\left(sinx + 2bcosx\right) + ce^{2bx}$
- $y^{2} = 2a\left(2bsinx + cosx\right) + ce^{2bx}$
The Correct Option is B
Solution and Explanation
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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