General solution of $\left(x+y\right)^{2} \frac{dy}{dx} = a^{2}, a \ne 0$ is (c is an arbitrary constant)
- $\frac{x}{a} = tan \frac{y}{a} + c$
- $tanxy = c$
- $tan \left(x + y\right) = c$
- $tan \frac{y+c}{a} = \frac{x+y}{a}$
The Correct Option is D
Solution and Explanation
$\Rightarrow\, \frac{z^{2}dz}{a^{2}+z^{2}} = dx$
Integrating
$\Rightarrow\,x+y-a\,tan^{-1} \frac{x+y}{a} = x+c, \quad\Rightarrow\,tan \left(\frac{y-c_{1}}{a}\right) = \frac{x+y}{a}$
$\Rightarrow\,tan\left(\frac{y+c}{a}\right) = \frac{x+y}{a}\quad\quad\left(c_{1} = -c\right)$
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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