The order of the differential equation of family of parabolas whose axes are parallel to y-axis is
A parabola is a U-shaped curve that is the graph of a quadratic function. It can open upward or downward depending on the sign of the leading coefficient. Parabolas are symmetric, with a vertex as the highest or lowest point. They have applications in physics, engineering, and geometry.
Class 12 Maths Chapter 9 on Differential Equations focuses on the relationship between a function and its derivatives. It covers methods for solving first order and higher-order differential equations. The chapter includes real-life applications like population growth and motion. This topic is important for CBSE Exam 2024-25 and its practical uses.
Share
The family of parabolas with axes parallel to the y-axis can be expressed as:
y = a x² + b x + c
which contains three arbitrary constants, all of them, namely a, b, and c.
A differential equation must therefore have a general solution containing three arbitrary constants. Three successive differentiations eliminate all the arbitrary constants. Indeed three successive differentiations of:
y = a x² + b x + c
After this gives:
y′ = 2a x + b
which twice gives:
y″ = 2a
and thrice gives:
y‴ = 0
So the differential equation that represents the family is:
y‴ = 0
This is a third order differential equation.
Thus, the order of the differential equation is 3.
Click here for more:
https://www.tiwariacademy.com/ncert-solutions/class-12/maths/#chapter-9