Class 12 Mathematics DIFFERENTIAL EQUATIONS Notes

Class 12 Mathematics DIFFERENTIAL EQUATIONS Notes
Differential Equations Notes | Nepal eNotes

Unit 17

Calculus

Class 12 Mathematics

Differential Equations

Class 12 Mathematics – Differential Equations Notes PDF

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Contents / Quick Navigation

1. Differential Equations2. Order & Degree 3. General & Particular Solutions4. Separable Variables 5. Homogeneous Equations6. First-Order Linear Equations 7. Exact Differential Equations8. Worked Examples 9. Method Selection10. Common Mistakes 11. Important Exam Questions12. One-Minute Revision

NEB / CDC Focus

This unit covers differential equations and their order and degree, with first-order first-degree equations solved by separable-variable, homogeneous, linear, and exact methods.

1. Differential Equations

Differential Equation

An equation involving an unknown function and one or more of its derivatives is called a differential equation.

Examples:

  • dy/dx=3x²
  • dy/dx+y=eˣ
  • d²y/dx²+4y=0
A Differential Equation Connects a Function with Its Derivatives y(x) dy/dxrate of change Equationrelates them
Figure 1: Solving a differential equation means finding the unknown function.

2. Order and Degree

Order

The order is the order of the highest derivative appearing in the differential equation.

Degree

After removing radicals and fractions involving derivatives, if the equation is polynomial in its derivatives, the degree is the power of the highest-order derivative.

EquationOrderDegree
dy/dx+x=011
(d²y/dx²)³+dy/dx=023
d³y/dx³+y=031

Degree is not defined in the usual school sense when the differential equation cannot be expressed as a polynomial in derivatives.

3. General and Particular Solutions

General Solution

A solution containing arbitrary constant(s) is called a general solution.

Particular Solution

A solution obtained after determining the constants from initial or boundary conditions is a particular solution.

Example

For dy/dx=2x,

y=x²+C.

If y=3 at x=1, then 3=1+C⇒C=2, so y=x²+2.

A Family of Solution Curves different CAn initial condition chooses one curve from the family.
Figure 2: Arbitrary constants create families of solution curves.

4. Differential Equations with Separable Variables

An equation is separable when it can be arranged so all y-terms occur with dy and all x-terms with dx.

dy/dx = f(x)g(y)dy/g(y)=f(x)dx∫dy/g(y)=∫f(x)dx.
Worked Example

Solve dy/dx=xy.

dy/y=x dx.

ln|y|=x²/2+C.

y=Ce^(x²/2).

Separation of Variables dy/dx=f(x)g(y) dy/g(y)=f(x)dxintegrate both sides
Figure 3: Separate the variables before integrating.

5. Homogeneous First-Order Differential Equations

A first-order equation of the form

dy/dx=F(y/x)

is homogeneous in this sense. Put

y=vx ⇒ dy/dx=v+x dv/dx.

The equation then becomes separable in v and x.

Worked Example

Solve dy/dx=1+y/x.

Let y=vx. Then dy/dx=v+x dv/dx.

v+x dv/dx=1+v ⇒ x dv/dx=1.

dv=dx/x ⇒ v=ln|x|+C.

y=x(ln|x|+C).

Do not confuse a homogeneous first-order equation dy/dx=F(y/x) with a homogeneous linear equation. The word “homogeneous” is used in more than one differential-equation context.

6. First-Order Linear Differential Equations

Standard form:

dy/dx+P(x)y=Q(x).

The integrating factor is

I.F.=e^(∫P(x)dx).

Then the solution is obtained from

y·I.F.=∫Q(x)·I.F. dx+C.
Worked Example

Solve dy/dx+y=eˣ.

P=1, so I.F.=eˣ.

yeˣ=∫e²ˣdx+C=e²ˣ/2+C.

y=eˣ/2+Ce⁻ˣ.

Linear Differential Equation Workflow Standardizey′+Py=Q Find I.F.e^(∫Pdx) Multiply equationby I.F. Integratesolve y
Figure 4: The integrating factor turns the left side into a product derivative.

7. Exact Differential Equations

Write the equation as

M(x,y)dx+N(x,y)dy=0.

It is exact if

∂M/∂y = ∂N/∂x.

Then there exists a potential function F(x,y) such that

dF=Mdx+Ndy,   solution: F(x,y)=C.

7.1 Solving an Exact Equation

  1. Check M_y=N_x.
  2. Integrate M with respect to x, treating y as constant.
  3. Add an unknown function φ(y).
  4. Differentiate the result with respect to y and compare with N.
  5. Find φ(y) and write F(x,y)=C.
Worked Example

Solve (2x+y)dx+(x+2y)dy=0.

M_y=1, N_x=1: exact.

Integrate M w.r.t. x:

F=x²+xy+φ(y).

F_y=x+φ′(y)=x+2y ⇒ φ′(y)=2y ⇒ φ=y².

x²+xy+y²=C.

Exact Equation Test Mdx+Ndy=0compute Mᵧ and Nₓ Mᵧ=Nₓ?Yes → find F(x,y)=C
Figure 5: Exactness is checked by comparing cross-partial derivatives.

8. More Worked Examples

Example 1: Order and Degree

(d²y/dx²)²+dy/dx+y=0 has order 2 and degree 2.

Example 2: Separable

dy/dx=x/y ⇒ y dy=x dx.

y²/2=x²/2+C ⇒ y²−x²=C.

Example 3: Homogeneous

dy/dx=y/x. Put y=vx: v+xv′=v⇒v′=0, so v=C, hence y=Cx.

Example 4: Linear

dy/dx+(2/x)y=x², x≠0.

I.F.=e^(∫2/x dx)=x².

yx²=∫x⁴dx+C=x⁵/5+C.

y=x³/5+C/x².

Example 5: Exact

2x dx+2y dy=0 is exact and integrates directly to x²+y²=C.

Which First-Order Method? Can separate x,y?Separable F(y/x)?Homogeneous y′+Py=Q?Linear Mᵧ=Nₓ?Exact
Figure 6: Method recognition saves time and prevents unnecessary algebra.

9. Method-Selection Strategy

  1. Identify order and degree.
  2. For first-order first-degree equations, try to separate variables first.
  3. If the right side is a function of y/x (or x/y), use the homogeneous substitution y=vx.
  4. If the equation fits y′+Py=Q, use the integrating factor.
  5. If written as Mdx+Ndy=0, test exactness.
  6. Apply any initial condition only after obtaining the general solution.
  7. Differentiate/substitute your final solution to check it.

10. Common Mistakes

Order vs degree

Order comes from the highest derivative; degree comes from its exponent after polynomial form.

Lost solution in separation

Dividing by y or another factor may discard a zero solution; check separately.

Homogeneous substitution

When y=vx, remember dy/dx=v+x dv/dx, not just x dv/dx.

Wrong integrating factor

For y′+Py=Q, I.F. is e^(∫Pdx).

Exactness test reversed

Compute ∂M/∂y and ∂N/∂x.

Missing constant

General solutions should contain the required arbitrary constant.

11. Formula Sheet

Separable: dy/g(y)=f(x)dx Homogeneous: y=vx, dy/dx=v+x dv/dx Linear: dy/dx+Py=Q, I.F.=e^(∫Pdx) Linear solution: y·I.F.=∫Q·I.F. dx+C Exact: Mdx+Ndy=0 with Mᵧ=Nₓ

12. Important Exam Questions

Short

  1. Define a differential equation, order and degree.
  2. Distinguish general and particular solutions.
  3. State the integrating factor for a first-order linear equation.
  4. State the condition for exactness.

Long / Numerical

  1. Solve a separable-variable equation.
  2. Solve a homogeneous first-order equation.
  3. Solve a first-order linear equation using integrating factor.
  4. Solve an exact differential equation.
  5. Use an initial condition to obtain a particular solution.

13. One-Minute Revision

  • A differential equation involves a function and derivatives.
  • Order = highest derivative order.
  • Degree = exponent of highest derivative after polynomial form.
  • Separable equations split x and y terms.
  • Homogeneous first-order: use y=vx.
  • Linear first-order: y′+Py=Q.
  • I.F. = e^(∫Pdx).
  • Exact equation: Mᵧ=Nₓ.
  • Initial conditions determine arbitrary constants.

14. Diagram Practice

  • Differential-equation concept map.
  • Family of solution curves.
  • Separation workflow.
  • Linear-equation integrating-factor workflow.
  • Exactness test.
  • Method-selection chart.

Also Visit

Class 12 Mathematics Notes Class 12 Notes Practical File Question Papers Important Questions
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