Chapter 9: Differential Equations - Board Exam Notes

Exam Weightage & Blueprint

Total: ~10-12 Marks

Differential Equations is one of the most important chapters for Board exams. It combines calculus concepts with practical problem-solving techniques.

Question Type	Marks	Frequency	Focus Topic
MCQ	1	Very High	Order & Degree, Basic concepts
Short Answer (2M)	2	High	Variable separable method
Long Answer (4M)	4	Very High	Homogeneous DE, Linear DE
Long Answer (6M)	6	Medium	Application problems

⏰ Last 24-Hour Checklist

Order: Highest order derivative
Degree: Power of highest order derivative (if polynomial)
Variable Separable: $\frac{dy}{h(y)} = g(x)dx$
Homogeneous: Put $y = vx$ or $x = vy$

Linear DE: $\frac{dy}{dx} + Py = Q$
I.F.: $e^{\int P dx}$
Solution: $y \cdot I.F. = \int Q \cdot I.F. \, dx + C$
Remember to substitute initial conditions

Basic Concepts ★★★★★

What is a Differential Equation?

Definition: An equation involving derivative(s) of the dependent variable with respect to independent variable(s) is called a differential equation.

Examples:
• $\frac{dy}{dx} + y = e^x$ (differential equation)
• $x^2 + 3x + 2 = 0$ (NOT a differential equation)

Order of a Differential Equation

Order: The order of the highest order derivative appearing in the equation.

Examples:
• $\frac{dy}{dx} = e^x$ → Order = 1
• $\frac{d^2y}{dx^2} + y = 0$ → Order = 2
• $\frac{d^3y}{dx^3} + x\frac{d^2y}{dx^2} = 0$ → Order = 3

Degree of a Differential Equation

Degree: When the differential equation is a polynomial in derivatives, the power (positive integer) of the highest order derivative is called the degree.

Important: Degree is NOT defined if the equation is not a polynomial in derivatives.

✓ Degree Defined

• $\left(\frac{d^2y}{dx^2}\right)^3 + 5\frac{dy}{dx} = 0$

Order = 2, Degree = 3

✗ Degree NOT Defined

• $\frac{dy}{dx} + \sin\left(\frac{dy}{dx}\right) = 0$

Not polynomial in $\frac{dy}{dx}$

Quick Identification:
1. Find the highest order derivative → That's the ORDER
2. Check if equation is polynomial in derivatives → If YES, find power of highest derivative → That's DEGREE
3. If NOT polynomial in derivatives → Degree is NOT defined

General & Particular Solutions ★★★★☆

General Solution: The solution containing arbitrary constants (as many as the order of DE).

Particular Solution: The solution obtained by giving specific values to arbitrary constants in the general solution.

Key Points:

A DE of order $n$ has $n$ arbitrary constants in its general solution
To find particular solution, use given initial conditions
Always verify your solution by substituting back in the original DE

Example: Verification

Q. Verify that $y = e^{-3x}$ is a solution of $\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0$

Solution:

Given: $y = e^{-3x}$

$\frac{dy}{dx} = -3e^{-3x}$

$\frac{d^2y}{dx^2} = 9e^{-3x}$

Substituting: $9e^{-3x} + (-3e^{-3x}) - 6e^{-3x} = 9e^{-3x} - 3e^{-3x} - 6e^{-3x} = 0$ ✓

Important Result:
Number of arbitrary constants in general solution = Order of DE
Number of arbitrary constants in particular solution = 0

Variable Separable Method ★★★★★

Method: If a DE can be written as $\frac{dy}{dx} = g(x) \cdot h(y)$, separate variables: $$\frac{1}{h(y)} dy = g(x) dx$$ Then integrate both sides.

Steps to Solve:
1. Check if DE can be written as $\frac{dy}{dx} = g(x) \cdot h(y)$
2. Separate: terms with $y$ on left, terms with $x$ on right
3. Integrate both sides: $\int \frac{1}{h(y)} dy = \int g(x) dx$
4. Add integration constant $C$ on one side only
5. Simplify to get the general solution

Example 1: Basic Variable Separable

Q. Solve: $\frac{dy}{dx} = \frac{1+x}{2-y}$ where $y \neq 2$

Step 1: Separate variables 1 Mark

$(2-y)dy = (1+x)dx$

Step 2: Integrate both sides 2 Marks

$\int (2-y) dy = \int (1+x) dx$

$2y - \frac{y^2}{2} = x + \frac{x^2}{2} + C$

Step 3: Simplify 1 Mark

$4y - y^2 = 2x + x^2 + 2C$

$x^2 + y^2 + 2x - 4y + K = 0$ (where $K = 2C$)

Example 2: Particular Solution

Q. Solve: $\frac{dy}{dx} = xy^2$ given $y = 1$ when $x = 0$

Step 1: Separate variables

$\frac{dy}{y^2} = x dx$

Step 2: Integrate

$\int y^{-2} dy = \int x dx$

$-\frac{1}{y} = \frac{x^2}{2} + C$

$y = \frac{-2}{x^2 + 2C}$

Step 3: Apply condition

When $x = 0, y = 1$: $1 = \frac{-2}{2C}$ → $C = -1$

Particular solution: $y = \frac{2}{2-x^2}$

Homogeneous Differential Equations 🔥🔥🔥

Homogeneous Function: A function $F(x,y)$ is homogeneous of degree $n$ if: $$F(\lambda x, \lambda y) = \lambda^n F(x, y)$$

Homogeneous DE: A DE of the form $\frac{dy}{dx} = F(x,y)$ where $F(x,y)$ is homogeneous of degree zero.

Equivalently: $\frac{dy}{dx} = g\left(\frac{y}{x}\right)$ or $\frac{dy}{dx} = h\left(\frac{x}{y}\right)$

Solution Method:
1. For $\frac{dy}{dx} = g\left(\frac{y}{x}\right)$: Put $y = vx$, then $\frac{dy}{dx} = v + x\frac{dv}{dx}$
2. For $\frac{dx}{dy} = h\left(\frac{x}{y}\right)$: Put $x = vy$, then $\frac{dx}{dy} = v + y\frac{dv}{dy}$
3. Substitute and separate variables in $v$ and $x$ (or $y$)
4. Integrate and replace $v$ by $\frac{y}{x}$ (or $\frac{x}{y}$)

Example: Homogeneous DE

Q. Solve: $(x-y)\frac{dy}{dx} = x + 2y$

Step 1: Check if homogeneous 0.5 Mark

$\frac{dy}{dx} = \frac{x+2y}{x-y} = \frac{1 + 2(y/x)}{1 - (y/x)} = g(y/x)$ → Homogeneous ✓

Step 2: Substitute $y = vx$ 1 Mark

$\frac{dy}{dx} = v + x\frac{dv}{dx}$

$v + x\frac{dv}{dx} = \frac{1 + 2v}{1 - v}$

Step 3: Separate variables 1.5 Marks

$x\frac{dv}{dx} = \frac{1 + 2v}{1-v} - v = \frac{1 + 2v + v^2 + v}{1-v} = \frac{v^2 + 3v + 1}{1-v}$

$\frac{1-v}{v^2 + 3v + 1} dv = -\frac{dx}{x}$

Step 4: Integrate 1 Mark

After integration and substituting $v = \frac{y}{x}$:

$\log|x^2 + y^2 + 2xy| - 2\sqrt{3}\tan^{-1}\left(\frac{2y+3x}{\sqrt{3}x}\right) = C$

Recognition Tips:
• If you can write $\frac{dy}{dx} = f(y/x)$ → Homogeneous
• Check: Replace $x$ with $\lambda x$ and $y$ with $\lambda y$. If $\lambda$ cancels → Homogeneous
• Common forms: $\frac{x^2 + y^2}{xy}$, $\frac{x+y}{x-y}$, etc.

Linear Differential Equations ★★★★★

Standard Form: A first-order linear differential equation has the form: $\frac{dy}{dx} + Py = Q$ where $P$ and $Q$ are functions of $x$ only (or constants).

Alternative form: $\frac{dx}{dy} + P_1x = Q_1$ where $P_1, Q_1$ are functions of $y$ only.

Solution Method:

Step 1: Write in standard form $\frac{dy}{dx} + Py = Q$

Step 2: Find Integrating Factor (I.F.): $I.F. = e^{\int P dx}$

Step 3: Solution is: $y \cdot (I.F.) = \int Q \cdot (I.F.) \, dx + C$

Memory Trick:
• $P$ is coefficient of $y$
• $Q$ is the term without $y$
• I.F. = $e^{\text{integral of P}}$
• Multiply entire equation by I.F., left side becomes $\frac{d}{dx}[y \cdot I.F.]$

Example 1: Linear DE

Q. Solve: $\frac{dy}{dx} - y = \cos x$

Step 1: Identify P and Q 0.5 Mark

Standard form: $\frac{dy}{dx} + (-1)y = \cos x$

$P = -1$, $Q = \cos x$

Step 2: Find I.F. 1 Mark

$I.F. = e^{\int (-1) dx} = e^{-x}$

Step 3: Apply formula 2 Marks

$y \cdot e^{-x} = \int \cos x \cdot e^{-x} dx$

Using integration by parts twice:

$y \cdot e^{-x} = \frac{e^{-x}(\sin x - \cos x)}{2} + C$

Step 4: Solve for y 0.5 Mark

$y = \frac{\sin x - \cos x}{2} + Ce^{x}$

Example 2: Alternative Form

Q. Solve: $x\frac{dy}{dx} + 2y = x^2$ (where $x \neq 0$)

Step 1: Standard form

$\frac{dy}{dx} + \frac{2}{x}y = x$

$P = \frac{2}{x}$, $Q = x$

Step 2: Find I.F.

$I.F. = e^{\int \frac{2}{x} dx} = e^{2\log|x|} = x^2$

Step 3: Solution

$y \cdot x^2 = \int x \cdot x^2 dx = \int x^3 dx = \frac{x^4}{4} + C$

$y = \frac{x^2}{4} + \frac{C}{x^2}$

Previous Year Questions (PYQs)

2023 (1 Mark MCQ): The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 = 5$ is:
(A) 1 (B) 2 (C) 3 (D) Not defined
Answer: (B) 2. The highest order derivative is $\frac{d^2y}{dx^2}$ with power 2.

2022 (4 Marks): Solve the differential equation: $(x+y)\frac{dy}{dx} = 1$
Solution: This is variable separable.
$(x+y)dy = dx$
Integrating: $xy + \frac{y^2}{2} = x + C$

2021 (4 Marks): Solve: $\frac{dy}{dx} + \frac{y}{x} = x^2$
Hint: This is linear DE with $P = \frac{1}{x}$, $Q = x^2$
$I.F. = e^{\int \frac{1}{x}dx} = x$
Solution: $y \cdot x = \int x^3 dx = \frac{x^4}{4} + C$

2020 (6 Marks): Find the particular solution of: $\frac{dy}{dx} = \frac{y(x-y)}{x(x+y)}$, given $y=1$ when $x=1$
Method: Homogeneous DE → Put $y = vx$

2019 (1 Mark MCQ): The integrating factor of $\frac{dy}{dx} + y\tan x = \sec x$ is:
(A) $\cos x$ (B) $\sec x$ (C) $e^{\tan x}$ (D) $\tan x$
Answer: (B) $\sec x$. Since $I.F. = e^{\int \tan x dx} = e^{\log|\sec x|} = \sec x$

Exam Strategy & Mistake Bank

Common Mistakes 🚨

Mistake 1: Confusing order and degree. Order = highest derivative, Degree = power of highest derivative.

Mistake 2: Forgetting to add constant $C$ after integration or adding it on both sides.

Mistake 3: In homogeneous DE, forgetting to replace $v$ by $\frac{y}{x}$ in final answer.

Mistake 4: Wrong I.F. calculation. Remember: $I.F. = e^{\int P dx}$, not $e^{P dx}$

Mistake 5: Not checking if degree is defined before stating it (must be polynomial in derivatives).

Scoring Tips 🏆

Tip 1: Always write the type of DE before solving (Variable Separable/Homogeneous/Linear).

Tip 2: For homogeneous DE, show the check: write it as $g(y/x)$ form.

Tip 3: In Linear DE, clearly write $P$, $Q$, and $I.F.$ - this shows your method.

Tip 4: For particular solution, substitute initial conditions and show calculation of $C$.

Tip 5: Practice Integration by Parts - it's crucial for Linear DE solutions.

Practice Problems (Self-Assessment)

Level 1: Basic (2 Marks Each)

Q1. Find order and degree: $\left(\frac{dy}{dx}\right)^3 + 2\frac{dy}{dx} = x$

Ans: Order = 1, Degree = 3

Q2. Solve: $\frac{dy}{dx} = e^{x+y}$

Hint: $e^{-y}dy = e^x dx$

Q3. Verify that $y = e^x + 1$ is solution of $y'' - y' = 0$

Level 2: Intermediate (4 Marks Each)

Q4. Solve: $(x^2+xy)dy = (x^2+y^2)dx$

Hint: Homogeneous DE

Q5. Solve: $\frac{dy}{dx} + 2y\tan x = \sin x$ where $0 \leq x < \frac{\pi}{2}$

Hint: Linear DE with $I.F. = \sec^2 x$

Level 3: Advanced (6 Marks Each)

Q6. Solve: $(1+x^2)\frac{dy}{dx} + 2xy = \frac{1}{1+x^2}$

Q7. Find equation of curve passing through $(1,1)$ whose differential equation is $x\frac{dy}{dx} = 2x^2 + y$

Formula Sheet (Must Remember!) 📝

Core Formulas

1. Order: Order of highest derivative

2. Degree: Power of highest derivative (if polynomial)

3. Variable Separable: $\int \frac{dy}{h(y)} = \int g(x) dx + C$

4. Homogeneous: Put $y = vx$, then $\frac{dy}{dx} = v + x\frac{dv}{dx}$

5. Linear DE: $\frac{dy}{dx} + Py = Q$

6. I.F.: $e^{\int P dx}$

7. Solution: $y \cdot (I.F.) = \int Q \cdot (I.F.) \, dx + C$

Quick Decision Tree:

Can you separate variables? → Use Variable Separable Method

Is it of form $\frac{dy}{dx} = g(y/x)$? → Use Homogeneous Method

Is it of form $\frac{dy}{dx} + Py = Q$? → Use Linear DE Method

Important Integration Results

1. $\int \frac{1}{x} dx = \log|x| + C$

2. $\int e^{ax} dx = \frac{e^{ax}}{a} + C$

3. $\int \sec^2 x dx = \tan x + C$

4. $\int \tan x dx = \log|\sec x| + C$

5. $e^{\log f(x)} = f(x)$