Let $b$ be a nonzero real number. Suppose $f : R \rightarrow R$ is a differentiable function such that $f(0)=1$. If the derivative $f^{\prime}$ of $f$ satisfies the equation
$f^{\prime}(x)=\frac{f(x)}{b^{2}+x^{2}}$
for all $x \in R$, then which of the following statements is/are TRUE?
$f^{\prime}(x)=\frac{f(x)}{b^{2}+x^{2}}$
for all $x \in R$, then which of the following statements is/are TRUE?
If \(b>0\), then $f$ is an increasing function
If \(b<0\), then $f$ is a decreasing function
- $f ( x ) f (- x )=1$ for all $x \in R$
- $f(x)-f(-x)=0$ for all $x \in R$
The Correct Option is A, C
Solution and Explanation
(A) If \(b>0\), then $f$ is an increasing function
(C) $f ( x ) f (- x )=1$ for all $x \in R$
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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