A differential equation is a mathematical equation involving a function and its derivatives. It describes how a quantity changes over time or space. Differential equations can be ordinary or partial depending on the number of independent variables and are widely used in various fields like physics engineering and biology.
Class 12 Maths Chapter 9 on Differential Equations deals with the relationship between a function and its derivatives. It covers methods for solving first-order and higher-order differential equations. Real-life applications such as motion population growth and decay are explored. This chapter is important for the CBSE Exam 2024-25 and practical problem-solving.
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Starting with the differential equation:
y (dy/dx) + x = C
Rearrange to get a derivative alone on one side
y (dy/dx) = C − x
Multiply by dx
y dy = (C − x) dx
Integrate both sides
∫ y dy = ∫ (C − x) dx
(1/2)y² = Cx − (1/2)x² + k, where k is the constant of integration
Multiply the whole equation by 2:
y² = 2Cx − x² + K, where K = 2k
Rewrite to bring together like terms:
x² + y² − 2Cx = −K
Complete the square for the x-terms:
(x − C)² + y² = C² − K
This is the equation of a circle with center at (C, 0) and radius √(C² − K).
Thus, the differential equation represents a family of circles.
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