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CBSE Class 12 · Mathematics

Differential Equations

Official NCERT chapter from Mathematics Part I–II (book code lemh1). ExamMaster notes are original teaching at CBSE Class 12 depth.

This lesson follows the official chapter “Differential Equations” in Mathematics Part I–II. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 12
  • Medium level
  • 8 concepts

1Basic Concepts

Basic concepts: an equation with y, x, and derivatives as taught. Order is the highest derivative. dy/dx=2x is first order, first degree as framed. A DE is a derivative-sentence, not an algebraic rearrange only.

Calling y=x^2 a DE because it came from one is a miss — that is a solution.

Figure. Order is the highest derivative that appears. Degree is the power of that highest derivative after the equation is polynomial in the derivatives. They are two different counts.

How it works

  1. Spot a derivative in the equationA DE.
  2. Name order (and degree if taught)The type.
  3. Keep a solution as a different wordy that fits.

2General and Particular Solutions of a

General solution has the taught number of arbitrary constants; a particular solution sits after a condition. For dy/dx=2x, y=x^2+C is general; y(0)=1 gives y=x^2+1. General is the family; particular is one member.

Dropping C and calling it general is a miss.

Figure. The general solution is a family of parallels y = x + C. A particular solution is one member; here y = x is the member through the origin.

How it works

  1. Integrate (or the taught method) and keep CGeneral.
  2. Use the condition to fix CParticular.
  3. Check by differentiating backThe fit.

Particular from dy/dx=2x

dy/dx=2x, y(0)=1. Find y.

  • Integratey=x^2+C
  • y(0)=1C=1
  • Particulary=x^2+1

Pro tip. Family, then the condition.

3Methods of Solving First Order, First Degree

First order, first degree methods (school): variable-separable, and homogeneous or linear if taught. dy/dx=2x separates as dy=2x dx. A method is a rewrite that lets you integrate both sides.

Treating every DE as separable without a check is a steal.

Figure. A first-order first-degree equation is solved by naming its type first: separable, homogeneous, or linear. Each path ends at a function y of x, not at a new type.

How it works

  1. Ask whether variables separate (or the taught form)The method.
  2. Integrate both sides and keep CThe solve.
  3. Name homogeneous/linear only if the lesson used themHonest extra.

4A definition is a test you can run

DE-word is a test: an order, a general/particular, or a separable solve. If you only say “equation”, you have a heading.

An algebra-sticker is not the test.

Figure. Run the order test on the written equation: the highest derivative present is the first derivative, so the order is 1. No second-derivative mark appears.

How it works

  1. Name order, family, or the methodThe object.
  2. Give the school sentenceThe test.
  3. Then the word has contentThe definition ran.

5Name the given before the unknown

The given is the DE and maybe a point. The unknown is y. Copy dy/dx=2x and y(0)=1 before you output y=x^2.

Using C=0 always is a silent skip of the condition.

Figure. The given is the initial pair (0, y0). Name that point before you pick C. The curve is the member of the family that is forced through the given point.

How it works

  1. Copy the DE and the conditionThe given.
  2. Name general then particularThe unknown.
  3. Then solveGiven first.

6One worked case is enough at this class

One worked case is enough: y=x^2+1. Do not stack four methods. The case teaches family-then-condition.

A poster of ten DEs is not more science.

Figure. Separable and already split: dy/dx = 2 integrates to y = 2x + C. The member through the origin is y = 2x. At data x = 0.4 the height is 0.8, which is twice the run.

How it works

  1. Take one first-order DEThe case.
  2. Integrate and fit CThe teach.
  3. Stop — one caseThis class.

7A miss: swapping the school name for the picture

A miss: swapping the school name for a picture. “Particular solution” is the member after a condition — a slope-field glow is a setting.

A pretty field without C-talk is a mute picture.

Figure. C in y = x + C slides the line without changing its slope. A steeper line is a different differential equation, not another value of C.

How it works

  1. Keep general/particular as the namesThe science.
  2. Use a picture only as a settingHonest.
  3. Refuse a swap of name for glowThe miss named.

8Check by the opposite action or the opposite test

Check by the opposite: differentiate your y and recover the DE; plug the condition. If y' is not the right-hand side, the solve failed.

A C that misses y(0)=1 is a failed particular.

Figure. The opposite test: differentiate the proposed solution. y = x squared plus 4 gives y prime = 2x, which is the differential equation. If the derivatives miss, the solution is wrong.

How it works

  1. Differentiate yShould match the DE.
  2. Plug the given pointThe condition.
  3. If either fails, redo CThe check.
dy/dx=2x, y(0)=1 gives
  1. y=x^2+1
  2. y=x^2
  3. y=2x

Family then C.

Notes

  • The official chapter title is “Differential Equations”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • dy/dx=2x => y=x^2+C
  • particular: use the given point

Recap

Hold these pegs from the official chapter “Differential Equations”. The wording is ExamMaster’s teaching, not a textbook recap.

Basic Concepts
Basic concepts: an equation with y, x, and derivatives as taught.
General and Particular Solutions of a
General solution has the taught number of arbitrary constants; a particular solution sits after a condition.
Methods of Solving First Order, First Degree
First order, first degree methods (school): variable-separable, and homogeneous or linear if taught.
A definition is a test you can run
DE-word is a test: an order, a general/particular, or a separable solve.
Name the given before the unknown
The given is the DE and maybe a point. The unknown is y. Copy dy/dx=2x and y(0)=1 before you output y=x^2.
One worked case is enough at this class
One worked case is enough: y=x^2+1. Do not stack four methods. The case teaches family-then-condition.

Practise Differential Equations

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