Question:
The slope at any point of a curve $y = f(x )$ is given by $\frac{d y}{d x}=3x^{2}$ and it passes through $(-1 ,1 )$ The equation of the curve is
The slope at any point of a curve $y = f(x )$ is given by $\frac{d y}{d x}=3x^{2}$ and it passes through $(-1 ,1 )$ The equation of the curve is
Updated On: Apr 23, 2024
- $y = x^{3}+2$
- $y=-x^{3}-2$
- $y=3x^{3}+4$
- $y=-x^{3}+2$
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Given, $\frac{dy}{dx}=3x^{2}$
$\Rightarrow dy=3x^{2}dx$
On integrating, we get
$y=\frac{3x^{3}}{3}+c $
$\Rightarrow y=x^{3}+c$
It passes through $(-1 ,1 )$
$\therefore 1=\left(-1\right)^{3}+c$
$\Rightarrow c=2$
$\therefore y=x^{3}+2$
$\Rightarrow dy=3x^{2}dx$
On integrating, we get
$y=\frac{3x^{3}}{3}+c $
$\Rightarrow y=x^{3}+c$
It passes through $(-1 ,1 )$
$\therefore 1=\left(-1\right)^{3}+c$
$\Rightarrow c=2$
$\therefore y=x^{3}+2$
Was this answer helpful?
0
0
Top Questions on Differential equations
- The general solution of the differential equation \( \frac{dy}{dx} = 3x^2 \) is:
- MHT CET - 2025
- Mathematics
- Differential equations
- General solution of the differential equation \[ \frac{dy}{dx} + y \tan x = \sec x \quad \text{is:} \]
- KCET - 2025
- Mathematics
- Differential equations
- If 'a' and 'b' are the order and degree respectively of the differentiable equation \[ \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^3 + x^4 = 0, \quad \text{then} \, a - b = \, \_ \_ \]
- KCET - 2025
- Mathematics
- 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
View More Questions
Questions Asked in WBJEE exam
- If 1000! = 3n × m, where m is an integer not divisible by 3, then n =?
- WBJEE - 2024
- Probability
- Chords AB CD of a circle intersect at right angle at the point P. If the lengths of AP, PB, CP, PD are 2, 6, 3, 4 units respectively, then the radius of the circle is:
- If the relation between the direction ratios of two lines in \(\mathbb{R}^3\) are given by \(l + m + n = 0\), \(2lm + 2mn - ln = 0\), then the angle between the lines is:
- WBJEE - 2024
- Vectors
- If \(a, b, c\) are distinct odd natural numbers, then the number of rational roots of the equation \(ax^2 + bx + c = 0\) is:
- WBJEE - 2024
- Quadratic Equation
- What force \(F\) is required to start moving this 10 kg block shown in the figure if it acts at an angle of \(60^\circ\) as shown? (\(\mu_s = 0.6\)).
- WBJEE - 2024
- laws of motion
View More Questions
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