Class 12 Mathematics — Chapter 3: DIFFERENTIAL EQUATIONS
Chapter 3: DIFFERENTIAL EQUATIONS is a chapter in Class 12 Mathematics (Part 2), part of the CBSE NCERT curriculum followed by over 25 million students across India. This chapter covers 8 topics including Order and Degree of a Differential Equation, General and Particular Solutions, Verification of Solutions. BrainWeave provides free AI-powered explanations — by voice or text, in Hindi or English — with no signup required.
What you'll learn
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▸Order and Degree of a Differential EquationCore conceptorderdegreehighest derivativepolynomialnot defined
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▸General and Particular SolutionsCore conceptgeneral solutionparticular solutionarbitrary constantsinitial conditionsfamily of curves
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▸Verification of Solutionsverify solutionsubstitutedifferentiatesatisfy equationLHS=RHS
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▸Formation of Differential EquationsCore conceptformationeliminatearbitrary constantsfamily of curvesdifferentiation
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▸Solving by Variables Separable MethodCore conceptvariables separableseparation of variablesintegrationfirst orderdy/dx = f(x)g(y)
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▸Solving Homogeneous Differential EquationsCore concepthomogeneousy/x formsubstitution y = vxdegreefirst order
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▸Solving Linear Differential EquationsCore conceptlinear differential equationintegrating factorI.F.dy/dx + Py = Qfirst order
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▸Applications of Differential Equationsapplicationword problempopulation growthradioactive decaymodeling
Chapter Summary
How to determine the order (the highest derivative present) and the degree (the highest power of the highest order derivative) of a differential equation, provided it is a polynomial in its derivatives. Students must also identify when the degree is not defined.
Understanding the distinction between a general solution, which contains arbitrary constants representing a family of curves, and a particular solution, which is obtained by assigning specific values to these constants to fit initial conditions.
The procedural skill of confirming whether a given function is a valid solution to a differential equation. This is done by differentiating the function as needed and substituting it and its derivatives back into the equation to see if it holds true.
The process of creating a differential equation from a given general solution (a family of curves). This involves differentiating the solution as many times as there are arbitrary constants and then algebraically eliminating those constants.
A fundamental technique for solving first-order, first-degree differential equations by separating all terms involving one variable to one side of the equation and all terms of the other variable to the other side, followed by integration.
Identifying and solving differential equations where dy/dx can be expressed as a function of y/x. The method involves the substitution y = vx, which transforms the equation into a variables separable form.
Solving first-order linear differential equations of the standard form dy/dx + P(x)y = Q(x). This involves finding the Integrating Factor (I.F.) and applying the standard solution formula y(I.F.) = ∫Q(x)(I.F.)dx + C.
Modeling and solving real-world problems using differential equations, such as population growth, radioactive decay, Newton's law of cooling, and mixing problems. This involves setting up the equation based on the problem statement and then solving it.
Practice Questions from this Chapter
Tap "Get Solution" on any question to ask our AI tutor.
- Explain differential equations simply. Get Solution →
- Give a real-world differential equation example. Get Solution →
- Show how derivatives connect to change. Get Solution →
- An equation involving the derivative of a dependent variable with respect to an independent variable is called what? Get Solution →
- What is the 'order' of a differential equation? Get Solution →
- For the degree of a differential equation to be defined, what must be true about the equation? Get Solution →
- A solution to a differential equation that is free from arbitrary constants is known as what? Get Solution →
- What is a solution to a differential equation that contains arbitrary constants called? Get Solution →
Did you know?
- 💡 Differential equations help predict everything from population growth to earthquake movements.
- 💡 Isaac Newton invented differential equations to understand how objects move and change speed.
- 💡 The way heat spreads through a metal bar is precisely described by a special differential equation.
- 💡 Doctors use differential equations to model how medicine travels through your body over time.
- 💡 Even the patterns on a leopard's skin can be explained using advanced forms of differential equations.
Frequently Asked Questions
How many topics are covered in this chapter?
This chapter covers 8 key topics: Order and Degree of a Differential Equation, General and Particular Solutions, Verification of Solutions, Formation of Differential Equations, Solving by Variables Separable Method, and more. The BrainWeave AI tutor explains each one with examples.
Is Chapter 3: DIFFERENTIAL EQUATIONS important for board exams?
Yes — Class 12 is a CBSE board exam year, and every NCERT chapter is part of the syllabus. Use BrainWeave's AI tutor to master this chapter, then practice with the auto-generated quizzes and mind maps.
Can I get NCERT solutions for this chapter in Hindi?
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