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: $y \cdot I.F. = \int Q \cdot I.F. \, dx + C$
Initial Conditions: Find $C$ using given $x, y$

Basic Concepts

Order & Degree

Order: The order of the highest order derivative appearing in the equation.
Degree: The power of the highest order derivative when the equation is 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 derivatives.

General & Particular Solutions

General Solution: Contains arbitrary constants equal to the order of DE.
Particular Solution: Obtained by giving specific values to constants (using initial conditions).

Methods of Solving DE

1. Variable Separable

If $\frac{dy}{dx} = g(x) \cdot h(y)$, separate as $\frac{1}{h(y)} dy = g(x) dx$ and integrate.

2. Homogeneous Differential Equations

Form: $\frac{dy}{dx} = g\left(\frac{y}{x}\right)$
Method: Put $y = vx \Rightarrow \frac{dy}{dx} = v + x\frac{dv}{dx}$.
Separate variables in $v$ and $x$, integrate, then put $v = y/x$.

3. Linear Differential Equations

Form: $\frac{dy}{dx} + Py = Q$ (where $P, Q$ are functions of $x$)
Integrating Factor (I.F.): $e^{\int P dx}$
Solution: $y \cdot (I.F.) = \int Q \cdot (I.F.) \, dx + C$

Solved Examples (Board Marking Scheme)

Q1. Solve: $\frac{dy}{dx} = \frac{1+x}{2-y}$ where $y \neq 2$. (2 Marks)

Step 1: Separate variables 1 Mark

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

Step 2: Integrate 1 Mark

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

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

Q2. Solve: $(x-y)\frac{dy}{dx} = x + 2y$. (4 Marks)

Step 1: Identify Homogeneous 1 Mark

$\frac{dy}{dx} = \frac{x+2y}{x-y}$. Put $y = vx \Rightarrow \frac{dy}{dx} = v + x\frac{dv}{dx}$.

Step 2: Separate Variables 1.5 Marks

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

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

Step 3: Integrate & Substitute 1.5 Marks

Integrate both sides and replace $v = y/x$.

Q3. Solve: $\frac{dy}{dx} - y = \cos x$. (4 Marks)

Step 1: Find I.F. 1 Mark

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

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

Step 2: Solution Formula 1 Mark

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

Step 3: Integrate RHS 2 Marks

Let $I = \int e^{-x} \cos x dx$. Use integration by parts twice.

$I = \frac{e^{-x}}{2}(\sin x - \cos x)$.

Final: $y = \frac{1}{2}(\sin x - \cos x) + Ce^x$.

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
Ans: (B). Highest order is 2, its power is 2.

2022 (4 Marks): Solve the differential equation: $(x+y)\frac{dy}{dx} = 1$.
Hint: Rewrite as $\frac{dx}{dy} = x+y \Rightarrow \frac{dx}{dy} - x = y$. This is Linear DE in $x$.
$P = -1, Q = y$. $I.F. = e^{-y}$. Solution: $x e^{-y} = \int y e^{-y} dy$.

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) $\tan x$
Ans: (B). $I.F. = e^{\int \tan x dx} = e^{\log|\sec x|} = \sec x$.

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.

Mistake 3: In homogeneous DE, forgetting to replace $v$ by $y/x$ in the final answer.

Scoring Tips

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

Tip 2: For Linear DE, clearly write $P$, $Q$, and $I.F.$ - this shows your method and earns partial marks.

Tip 3: For particular solution, substitute initial conditions and show calculation of $C$.

Practice Problems (Self-Assessment)

Level 1: Basic (2 Marks Each)

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

Answer: Order = 1, Degree = 3.

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

Answer: $e^{-y} dy = e^x dx \Rightarrow -e^{-y} = e^x + C \Rightarrow e^x + e^{-y} = K$.

Level 2: Intermediate (4 Marks Each)

Q3. Solve: $\frac{dy}{dx} + \frac{y}{x} = x^2$.

Hint: Linear DE. $P = 1/x, Q = x^2$. $I.F. = x$.
Solution: $yx = \int x^3 dx = x^4/4 + C$.

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

Hint: Homogeneous DE. Put $y=vx$.