Doubtclip is an AI tutor that turns any homework or exam doubt into a step-by-step whiteboard video lesson with diagrams, formulas, and voice explanation.

Who is the founder of Doubtclip?

Vinay Malav is the founder of Doubtclip (Founder, Doubtclip, based in Kota, Rajasthan, India). More: https://www.doubtclip.com/about

Which subjects and exams does it support?

It supports Physics, Chemistry, Mathematics/Calculus and related STEM doubts for school boards, JEE, NEET, and global high-school / undergrad practice.

How do I get a video solution?

Upload a photo of the question or paste the text, sign in, and Doubtclip generates a whiteboard-style explanation video you can watch and share.

Is Doubtclip available worldwide?

Yes. Doubtclip is a web app for students globally. Lessons are available in English and Hinglish.

What is the primary governing principle of Differential Equations: Linear & Variable Separable?

Fundamental thermodynamic and energy minimization principles. Grounded directly in conservation laws and quantum-mechanical principles.

In competitive examinations (JEE/NEET), what is the most critical pitfall in Differential Equations: Linear & Variable Separable?

All of the above. Students frequently lose marks on sign conventions, units, and temperature-dependent behavior.

Mathematics · Class 12 / JEE Main & Advanced

Differential Equations: Linear & Variable Separable

Master ordinary differential equations (ODEs): Order and degree, variable separable method, homogeneous equations, and first-order linear ODEs with Integrating Factor ( $e^{\int P dx}$ ).

11. Visual Overview & Intuitive Discovery: Differential Equations: Linear & Variable Separable

🔍 Clickable Interactive Study Prompts (Click to explore or jump to section):

Clickable

AI Deep Dive 🪄

### Intuitive Foundations & Real-World Reality

In 1676, Sir Isaac Newton wrote in an anagram: "Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa" (It is useful to solve differential equations). Differential equations translate the dynamic laws of nature—gravitation, fluid velocity, heat dissipation, and radioactive decay—into predictive equations that govern everything from epidemic spread to financial markets.

Differential Equations: Linear & Variable Separable Conceptual Framework: Core Anchor 1, Core Anchor 2, Core Anchor 3, Core Anchor 4 — Differential Equations: Linear & Variable Separable Conceptual Framework
Master ordinary differential equations (ODEs): Order and degree, variable separable method, homogeneous equations, and first-order linear ODEs with Integrating Factor ($e^{\int P dx}$).
1Core Anchor 1 — Derive the Integrating Factor Formula (IF = e^∫P dx)
2Core Anchor 2 — Variable Separable vs Homogeneous ODE Methods
3Core Anchor 3 — Newton's Law of Cooling Differential Solutions
4Core Anchor 4 — Orthogonal Trajectories and Population Growth Models

22. Concept & Decision Flowchart: First Order ODE Solution Strategy Flowchart

First Order ODE Solution Strategy Flowchart

1
1. Check Variable Separability
Can it be written as g(y)dy = f(x)dx? If yes, integrate both sides directly: ∫g(y)dy = ∫f(x)dx + C.
2
2. Check Homogeneity
If dy/dx = F(y/x), substitute y = vx (dy/dx = v + x dv/dx) to reduce into variable separable form.
3
3. Identify Linear Form (dy/dx + Py = Q)
P and Q must be functions of x only (or constants). Compute Integrating Factor IF = e^(∫P dx).
4
4. General Linear Solution
Multiply through by IF: y × (IF) = ∫ [Q × (IF)] dx + C.

33. Step-by-Step Derivation: The Integrating Factor for First Order Linear ODEs

Derivation — Step-by-Step Derivation: The Integrating Factor for First Order Linear ODEs

1
$\frac{dy}{dx} + P(x) y = Q(x)$
Where P(x) and Q(x) are continuous functions of independent variable x.
2
$\mu(x) \frac{dy}{dx} + \mu(x) P(x) y = \mu(x) Q(x)$
We demand that the left-hand side matches the exact product rule derivative d/dx [μ(x) y].
3
$\frac{d}{dx}[\mu(x) y] = \mu(x) \frac{dy}{dx} + \frac{d\mu}{dx} y \implies \frac{d\mu}{dx} = \mu(x) P(x)$
Comparing coefficients of y yields a separable differential equation for μ(x).
4
$\int \frac{d\mu}{\mu} = \int P(x)\,dx \implies \ln \mu = \int P(x)\,dx \implies \mu(x) = e^{\int P(x)\,dx}$
Integrating both sides of the exact differential yields the master solution: y × e^(∫P dx) = ∫ Q e^(∫P dx) dx + C.

44. Real-World Case Study & Industrial Application

### Epidemiology & Virology: The SIR Differential Model of Infection

The Kermack-McKendrick SIR model uses a coupled system of differential equations: dS/dt = -βSI (susceptible depletion), dI/dt = βSI - γI (infectious spread and recovery), and dR/dt = γI. Solving these differential equations allows epidemiologists to calculate the Basic Reproduction Number R0 = β / γ and predict the exact peak of viral pandemics to schedule vaccine rollouts.

Common mistake: Common Exam Trap & Misconception: Order and Degree are frequently confused! Order is the highest derivative present in the equation (e.g. d²y/dx² has order 2). Degree is the exponent of the highest derivative AFTER removing fractional and radical powers. If dy/dx appears inside a sin(dy/dx) or e^(dy/dx), the degree is NOT DEFINED!

55. High-Yield Reference Data & Solved Problems

High-Yield Comparison & Key Constants

Differential Equation Type	Standard Algebraic Form	Solution Technique
Variable Separable	$dy/dx = f(x) g(y)$	∫ [1/g(y)] dy = ∫ f(x) dx + C
Homogeneous ODE	$dy/dx = f(x, y) \text{where}\ f(tx, ty) = f(x, y)$	$\text{Substitute}\ y = v x; dy/dx = v + x(dv/dx)$
Linear ODE in y	$dy/dx + P(x) y = Q(x)$	y (IF) = ∫ Q(IF) dx + C; IF = e^(∫P dx)
Linear ODE in x	$dx/dy + P(y) x = Q(y)$	x (IF) = ∫ Q(IF) dy + C; IF = e^(∫P dy)

Variable Separable
Standard Algebraic Form
$dy/dx = f(x) g(y)$
Solution Technique
∫ [1/g(y)] dy = ∫ f(x) dx + C
Homogeneous ODE
Standard Algebraic Form
$dy/dx = f(x, y) \text{where}\ f(tx, ty) = f(x, y)$
Solution Technique
$\text{Substitute}\ y = v x; dy/dx = v + x(dv/dx)$
Linear ODE in y
Standard Algebraic Form
$dy/dx + P(x) y = Q(x)$
Solution Technique
y (IF) = ∫ Q(IF) dx + C; IF = e^(∫P dx)
Linear ODE in x
Standard Algebraic Form
$dx/dy + P(y) x = Q(y)$
Solution Technique
x (IF) = ∫ Q(IF) dy + C; IF = e^(∫P dy)

Exam question · JEE Main, Advanced & NEET Standard

Solve the differential equation: x (dy/dx) + 2y = x² (x > 0) with initial condition y(1) = 1.

Remember

Differential Equations: Linear & Variable Separable: Foundational law governing physical, chemical, and biological phenomena.
Mastered derivation: The Integrating Factor for First Order Linear ODEs.
Avoided classic trap: Order and Degree are frequently confused! Order is the highest derivative present in the equation (e.g. d²y/dx...
Real-world relevance: Proven clinical, aerospace, and industrial significance.
High-yield exam rule: Always verify dimensional consistency and physical boundary conditions.

Check yourself

Question 1
What is the primary governing principle of Differential Equations: Linear & Variable Separable?
Question 2
In competitive examinations (JEE/NEET), what is the most critical pitfall in Differential Equations: Linear & Variable Separable?