Differential Equations

Class 12 · Mathematics

4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.

Order, Degree and Solutions

Part of: Basics and Separable Equations

The order is the highest derivative present, while the degree is its power once the equation is a polynomial in derivatives; transcendental wrappings leave degree undefined. Solutions are functions satisfying the equation, general with constants and particular with fixed values.

Why it matters. Order and degree classify every equation, and verification is the fastest correctness check.

Formulas

Order is the highest derivative present
Degree needs polynomial form in derivatives
Second-order general solutions carry two constants

Key words

Order: : The highest derivative present in the equation.
Degree: : The power of the highest derivative, when polynomial.
General solution: : A solution carrying arbitrary constants.
Particular solution: : A solution with constants fixed to values.

Try it in two minutes

Circle the highest derivative for order.
Check polynomial form before reading degree.
Plug a candidate solution back in.

Try at home. Find order and degree of (y′′)² + y′ + y = 0.

Variable Separable Equations

Part of: Basics and Separable Equations

Separable equations split so each side owns one variable, then both sides integrate independently. The constant rides along, and a given point pins it down into a particular solution.

Why it matters. Separation is the first method tried on any first-order equation.

Formulas

Split variables to opposite sides first
Integrate each side independently
Given points fix the constant

Key words

Separable equation: : An equation splittable so each side owns one variable.
General solution: : The answer family before fixing constants.
Particular solution: : The member passing through the given point.
Initial point: : The given pair fixing the constant.

Try it in two minutes

Divide and multiply until variables split sides.
Integrate each side alone.
Pin the constant with the given point.

Try at home. Solve dy/dx = 4x³ with y(1) = 2.

Homogeneous Equations

Part of: Homogeneous and Linear Equations

Homogeneous equations balance every term to one degree, or write dy/dx as a function of y/x alone. The substitution y = vx melts them into separable form in v and x, and points fix the constant after back-substitution.

Why it matters. Many symmetric rate problems arrive homogeneous, and y = vx is the master key.

Formulas

Equal degree in every term signals homogeneous
Put y = vx, then dy/dx is v plus x dv/dx
Separated v and x integrate independently

Key words

Homogeneous function: : A function scaling cleanly with degree n.
Degree of homogeneity: : The common power of all terms.
Substitution y = vx: : The costume change making homogeneous separable.
Back-substitution: : Returning v to y/x after solving.

Try it in two minutes

Check every term shares one degree.
Put y = vx and rewrite dy/dx.
Separate v and x, then return to y.

Try at home. Test (x³ + y³)dx − xy²dy = 0 for homogeneity and name the degree.

Linear Equations

Part of: Homogeneous and Linear Equations

Linear equations keep y and its derivative to power one: dy/dx + Py = Q, or the mirror dx/dy + Rx = S. Multiplying by e to the integral of P collapses the left side into a single derivative that integrates directly.

Why it matters. Growth, mixing and circuit problems all linearise into this form.

Formulas

Standard form leads with dy/dx plus Py
Integrating factor is e to the integral of P
Collapsed side integrates in one step

Key words

Linear equation: : An equation with y and derivative to power one.
Integrating factor: : The multiplier collapsing the left side.
Standard form: : dy/dx + Py = Q with leading coefficient one.
Mirror linear form: : dx/dy + Rx = S, linear in x.

Try it in two minutes

Divide into standard form first.
Build e to the integral of P.
Collapse, integrate, and divide back.

Try at home. Solve dy/dx + 4y = 8 with y(0) = 1.

What your child will practise

Homogeneous Equations: Candy Quick Fire (Quick Fire)
Homogeneous Equations: Space Puzzle (Clue Puzzle)
Linear Equations: Ocean Puzzle (Clue Puzzle)
Order, Degree and Solutions: Space Quest (Flip and Match)
Variable Separable Equations: Ocean Quick Fire (Quick Fire)

Quiz: Homogeneous Equations (8 questions)
Quiz: Linear Equations (8 questions)
Quiz: Order, Degree and Solutions (8 questions)
Quiz: Variable Separable Equations (8 questions)

Differential Equations

Class 12 · Mathematics

4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.

Order, Degree and Solutions

Part of: Basics and Separable Equations

Why it matters. Order and degree classify every equation, and verification is the fastest correctness check.

Formulas

Order is the highest derivative present
Degree needs polynomial form in derivatives
Second-order general solutions carry two constants

Key words

Order: : The highest derivative present in the equation.
Degree: : The power of the highest derivative, when polynomial.
General solution: : A solution carrying arbitrary constants.
Particular solution: : A solution with constants fixed to values.

Try it in two minutes

Circle the highest derivative for order.
Check polynomial form before reading degree.
Plug a candidate solution back in.

Try at home. Find order and degree of (y′′)² + y′ + y = 0.

Variable Separable Equations

Part of: Basics and Separable Equations

Separable equations split so each side owns one variable, then both sides integrate independently. The constant rides along, and a given point pins it down into a particular solution.

Why it matters. Separation is the first method tried on any first-order equation.

Formulas

Split variables to opposite sides first
Integrate each side independently
Given points fix the constant

Key words

Separable equation: : An equation splittable so each side owns one variable.
General solution: : The answer family before fixing constants.
Particular solution: : The member passing through the given point.
Initial point: : The given pair fixing the constant.

Try it in two minutes

Divide and multiply until variables split sides.
Integrate each side alone.
Pin the constant with the given point.

Try at home. Solve dy/dx = 4x³ with y(1) = 2.

Homogeneous Equations

Part of: Homogeneous and Linear Equations

Why it matters. Many symmetric rate problems arrive homogeneous, and y = vx is the master key.

Formulas

Equal degree in every term signals homogeneous
Put y = vx, then dy/dx is v plus x dv/dx
Separated v and x integrate independently

Key words

Homogeneous function: : A function scaling cleanly with degree n.
Degree of homogeneity: : The common power of all terms.
Substitution y = vx: : The costume change making homogeneous separable.
Back-substitution: : Returning v to y/x after solving.

Try it in two minutes

Check every term shares one degree.
Put y = vx and rewrite dy/dx.
Separate v and x, then return to y.

Try at home. Test (x³ + y³)dx − xy²dy = 0 for homogeneity and name the degree.

Linear Equations

Part of: Homogeneous and Linear Equations

Why it matters. Growth, mixing and circuit problems all linearise into this form.

Formulas

Standard form leads with dy/dx plus Py
Integrating factor is e to the integral of P
Collapsed side integrates in one step

Key words

Linear equation: : An equation with y and derivative to power one.
Integrating factor: : The multiplier collapsing the left side.
Standard form: : dy/dx + Py = Q with leading coefficient one.
Mirror linear form: : dx/dy + Rx = S, linear in x.

Try it in two minutes

Divide into standard form first.
Build e to the integral of P.
Collapse, integrate, and divide back.

Try at home. Solve dy/dx + 4y = 8 with y(0) = 1.

What your child will practise

Homogeneous Equations: Candy Quick Fire (Quick Fire)
Homogeneous Equations: Space Puzzle (Clue Puzzle)
Linear Equations: Ocean Puzzle (Clue Puzzle)
Order, Degree and Solutions: Space Quest (Flip and Match)
Variable Separable Equations: Ocean Quick Fire (Quick Fire)

Quiz: Homogeneous Equations (8 questions)
Quiz: Linear Equations (8 questions)
Quiz: Order, Degree and Solutions (8 questions)
Quiz: Variable Separable Equations (8 questions)