The family of curves y=easin x, where a is an arbitrary constant, is represented by the differential equation
- ylog y=tanx \(\frac{dy}{dx}\)
- ylog x=cotx \(\frac{dy}{dx}\)
- log y=tanx \(\frac{dy}{dx}\)
- log y=cotx \(\frac{dy}{dx}\)
The Correct Option is A
Solution and Explanation
The correct answer is option (A): ylog y=tanx \(\frac{dy}{dx}\)
\(y=e^{a\,sin\,x}\)
\(\Rightarrow log\,y=a\,sin\,x....(i)\)
Differentiating w.r.t \(x\), we get
\(\frac{1}{y}.\frac{dy}{dx}=a\,cos\,x\)
\(\Rightarrow a=\frac{1}{y\,cos\,x}\frac{dy}{dx}\)
Putting the value of a in (I), we get,
ylog y=tanx \(\frac{dy}{dx}\)
Top Questions on Differential equations
- Let \( y = f(x) \) be the solution of the differential equation\[\frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^6 + 4x}{\sqrt{1 - x^2}}, \quad -1 < x < 1\] such that \( f(0) = 0 \). If \[6 \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha\] then \( \alpha^2 \) is equal to __________.
- JEE Main - 2025
- Mathematics
- Differential equations
- If \( x = f(y) \) is the solution of the differential equation \[ (1 + y^2) + (x - 2e^{\tan^{-1}y}) \frac{dy}{dx} = 0, \quad y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right), \] with \( f(0) = 1 \), then \( f\left( \frac{1}{\sqrt{3}} \right) \) is equal to:
- JEE Main - 2025
- Mathematics
- Differential equations
- Let \( f(x) \) be a real differentiable function such that \( f(0) = 1 \) and \( f(x + y) = f(x)f'(y) + f'(x)f(y) \) for all \( x, y \in \mathbb{R} \). Then \( \sum_{n=1}^{100} \log_e f(n) \) is equal to :
- JEE Main - 2025
- Mathematics
- Differential equations
- Let \( x = x(y) \) be the solution of the differential equation \( y^2 dx + \left( x - \frac{1}{y} \right) dy = 0 \). If \( x(1) = 1 \), then \( x\left( \frac{1}{2} \right) \) is :
- JEE Main - 2025
- Mathematics
- Differential equations
Let \( f : \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f(x + y) = f(x) f(y) \) for all \( x, y \in \mathbb{R} \). If \( f'(0) = 4a \) and \( f \) satisfies \( f''(x) - 3a f'(x) - f(x) = 0 \), where \( a > 0 \), then the area of the region R = {(x, y) | 0 \(\leq\) y \(\leq\) f(ax), 0 \(\leq\) x \(\leq\) 2\ is :
- JEE Main - 2025
- Mathematics
- Differential equations
Questions Asked in WBJEE exam
- Let \(f : \mathbb{R} \to \mathbb{R}\) be given by \(f(x) = |x^2 - 1|\), then:
- \(f(x) = \cos x - 1 + \frac{x^2}{2!}, \, x \in \mathbb{R}\)
Then \(f(x)\) is: - Let \(f : \mathbb{R} \to \mathbb{R}\) be a differentiable function and \(f(1) = 4\). Then the value of
\[ \lim_{x \to 1} \int_{4}^{f(x)} \frac{2t}{x - 1} \, dt \]- WBJEE - 2024
- Integration
- If \(\int \frac{\log(x + \sqrt{1 + x^2})}{1 + x^2} \, dx = f(g(x)) + c\), then:
- WBJEE - 2024
- Integration
- The equation \(2x^5 + 5x = 3x^3 + 4x^4\) has:
- WBJEE - 2024
- Quadratic Equation
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