Class 12 Maths - MAHARASHTRA
Differential Equations
The Differential Equations chapter in Class 12 Mathematics under the Maharashtra (MSBSHSE) board builds upon calculus to study equations involving derivatives. Students learn to determine the order and degree of differential equations, form them from given conditions, and solve first-order, first-degree differential equations using methods like variable separable, homogeneous, and linear differential equations. Real-world applications such as population growth, radioactive decay, and physics problems are also explored. This chapter is vital for the HSC board exams, carrying a significant weightage of marks with a mix of direct numerical problems and word problems.
Start Learning FreeKey Concepts
Order and Degree of a Differential Equation
The order is the highest derivative present in the equation, and the degree is the highest power of that highest derivative, provided the equation is a polynomial in its derivatives.
General and Particular Solution
A solution containing arbitrary constants equal to the order of the differential equation is the general solution, while a solution obtained by assigning specific values to these constants to satisfy given conditions is a particular solution.
Variable Separable Method
A technique used to solve first-order differential equations by algebraically rearranging terms so that all expressions containing 'x' and 'dx' are on one side, and 'y' and 'dy' are on the other, followed by integration.
Homogeneous Differential Equation
A differential equation where every term is of the same degree, solved by substituting y = vx and dy/v = v dx + x dv to reduce it to a variable separable form.
Linear Differential Equation
An equation of the form dy/dx + Py = Q (where P and Q are functions of x only), solved using an integrating factor (I.F.) given by e^(∫P dx).
Important Formulas
Board Exam Info
In the Maharashtra (MSBSHSE) Class 12 Mathematics board exam, the Differential Equations chapter typically carries around 6 to 8 marks (up to 10 marks with options). Questions frequently include finding order and degree (1-2 marks), solving variable separable or homogeneous equations (3-4 marks), and solving linear differential equations or application-based word problems (4 marks).
Frequently Asked Questions
How do we know if a differential equation is homogeneous?
An equation f(x, y) dy/dx = g(x, y) is homogeneous if every term in both functions has the same total degree, or equivalently, if replacing x and y with kx and ky results in k^n * f(x, y).
What is an Integrating Factor (I.F.) and when is it used?
An Integrating Factor is a special multiplier used exclusively to solve first-order linear differential equations that are not directly separable, transforming the non-exact or non-integrable left side into an exact derivative.
When should we add the constant 'c' during integration?
The constant of integration 'c' must be added as soon as you perform the final integration step on both sides of the separated equation to write the general solution.
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