Question:
The differential equation which represents the family of curves $y=c_1e^{c_2x}$, where $c_1$ and $c_2$ are arbitrary constants is
The differential equation which represents the family of curves $y=c_1e^{c_2x}$, where $c_1$ and $c_2$ are arbitrary constants is
Updated On: Jul 5, 2022
- $y '=y^2$
- $y ''=y' y$
- $yy ''=y'$
- $yy ''=(y')^2$
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The Correct Option is D
Solution and Explanation
$y=c_{1}e^{c_{2}x}\,...\left(1\right)$
$y '=c_{2}c_{1}e^{c_{2}x}$
$y '=c_{2}y \,...\left(2\right)$
$y ''=c_{2}y '$
From $\left(2\right)$
$c_{2}=\frac{y '}{y}$
So, $y "=\frac{\left(y '\right)^{2}}{y} \Rightarrow yy "=\left(y '\right)^{2}$
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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