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Calculus

Differential Equations

Mathematics Core
20 min read
IAT Core Framework
High Yield

1. Core Concept

A Differential Equation (DE) is an equation involving an unknown function and its derivatives. It represents a physical relationship between a quantity and its rate of change. Solving a DE means finding the entire family of curves that satisfy this relationship.

Geometric Meaning of Solution

The solution of a differential equation represents a family of curves.

Each value of constant C gives a different curve.

2. Key Formulas & Concepts

  • Order: The highest order derivative present in the equation.
  • Degree: The power of the highest order derivative (applicable only when the DE can be expressed as a polynomial in its derivatives).

Variable Separable

If you can group all y's with dy and x's with dx:

If

dydx
= f(x)g(y), then ∫
dyg(y)
= ∫ f(x) dx + C

Homogeneous DE

If

dydx
=
f(x,y)g(x,y)
where both are homogeneous functions of the same degree:

Substitute: y = vx   and  
dydx
= v + x
dvdx

Form 1:
dydx
+ Py = Q (where P, Q are functions of x)
Integrating Factor (IF): e∫ P dx
Solution: y · (IF) = ∫ (Q · IF) dx + C
Form 2:
dxdy
+ Px = Q (where P, Q are functions of y)
Integrating Factor (IF): e∫ P dy
Solution: x · (IF) = ∫ (Q · IF) dy + C
Linear Differential Equation (LDE): Most frequently tested concept in the exam.

Exact Differential Equation

If equation is of form:
M(x,y)dx + N(x,y)dy = 0

Condition:
∂M/∂y = ∂N/∂x

Solution:
Integrate M w.r.t x and N w.r.t y and combine → result = C

3. Important Results

  • Formation of DE: To form a DE from a family of curves, differentiate the equation n times (where n is the exact number of arbitrary constants) and eliminate the constants entirely.
  • General vs. Particular Solution:
    • General: Contains arbitrary constants (matches the order of the DE).
    • Particular: Constants are evaluated using given boundary/initial conditions.

    • Applying Initial Conditions

    After finding general solution, substitute given values of x and y to find constant C.

    Example: If y = Cex and y(0) = 2 → C = 2 → y = 2ex

  • Orthogonal Trajectories: To find a family of curves that are perfectly perpendicular to the given family, replace
    dydx
    with -
    dxdy
     in the DE and solve.

4. Conceptual Insights

  • Order vs. Degree: Order is always defined, but degree is NOT defined if the derivative is trapped inside a non-polynomial function like sin(y′), ey′, or ln(y′).
  • Integrating Factor (IF) Magic: The IF is a brilliant mathematical multiplier that transforms a non-separable equation into a perfect, easily integrable derivative:
    ddx
    (y · IF).
  • Visual Intuition: A first-order DE fundamentally defines a slope field. The solution curve is simply a path that follows the arrows of the slope field at every single coordinate point.

5. Common Mistakes

  • Degree Trap: Finding the degree before removing radicals (square roots) or fractional powers. You must simplify and rationalize the equation first!
  • The "ln" in IF: Remember log properties! eln(f(x)) = f(x). Do not leave it as eln x. Also, e-ln x =
    1x
    . Don't forget the negative sign flips it!
  • Constants of Integration: Forgetting +C in the middle of a multi-step integration. In DEs, +C can totally change the form of the final function if multiplied by other terms later.
  • Wrong LDE Form: Using the dy/dx formula for an equation that is clearly dx/dy. Always check whether P and Q depend strictly on x or strictly on y first.

6. Example Application Insights

Growth/Decay Models

dydt
= ky &implies; y = C ekt. If k > 0, it represents exponential growth. If k < 0, it represents radioactive decay.

Family of Circles

Consider circles at the origin: x2 + y2 = r2. Differentiating gives:
2x + 2y·y′ = 0 &implies; y′ = -x/y.
Notice the constant r is cleanly eliminated in one step, making it a 1st order DE.

IF Calculation Example

If

dydx
+
2x
y = x, then:

IF = e∫

2x
dx = e2 ln x = eln(x2) = x2.

7. IAT Exam Focus Points

How to Identify Type Quickly

  • dy/dx = f(x)g(y) → Variable separable
  • dy/dx = f(y/x) → Homogeneous → use y = vx
  • dy/dx + Py = Q → Linear DE → use IF
  • Mdx + Ndy = 0 → Check exactness
  • Order and Degree: Almost guaranteed as a 1-mark or conceptual part. Actively check for "Degree is not defined" trick questions.
  • Linear DE (dy/dx + Py = Q): The most frequently tested type. Practice finding the Integrating Factor quickly for P = tan(x), sec(x), or 1/x.
  • Variable Separable with Logarithms: Be ready to use exponential ex to remove ln functions efficiently after integrating both sides.
  • Formation Shortcut: Knowing that equations of the form y = A cos(x) + B sin(x) always lead directly to the DE y″ + y = 0. (Two constants → 2nd order).
  • Substitution in Homogeneous: Recognising forms like
    dydx
    = f(y/x). You must substitute y = vx immediately to unlock the solution.

8. Practice Mock Test

Ready to test your knowledge?

Take a quick 15-question assessment specifically designed for Differential Equations. Challenge yourself with IAT-level questions.

Start Practice Mock

End of Chapter

Differential Equations

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