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🚀 COMPLETE PREPARATION LOOP: LEARN → PRACTICE → ANALYZE

Differential Equations Preparation Hub

Master Differential Equations with coaching-grade theory notes, verified video lectures, and 10 interactive MCQs with instant mistake analysis. Continue with 500+ adaptive questions inside the Exam Sprinter app.

📘 Theory Notes ✍️ Interactive Practice ⏱️ CBT Simulator

What is Differential Equations?

Differential Equations covers the essential principles and concepts required for JEE Main.

💡 Why Study Differential Equations?

Highly important, frequently tested in JEE Main.

Parameter	Details / Relevance
Target Exam	JEE Main
Subject Category	Mathematics
Estimated Study Duration	10 Hours
Expected Questions	2 Questions
Difficulty Index	Easy
Interactive Solved MCQs	10 Questions with AI Diagnostics
Adaptive App Practice	500+ Questions & Real-Time AI Tutor

⚠️ Common Pitfalls to Avoid

Rushing through mathematical derivations without checking boundary conditions and signs.
Guessing options when under time pressure rather than systematically eliminating choices.
Confusing intermediate algebraic steps with the final required answer value.

✓ COACHING-GRADE CLASSROOM NOTES

Differential Equations Theory & Derivations

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Differential Equations

Overview

This is the fully verified JEE Main content for Differential Equations generated via the 10-stage premium pipeline.

Subtopics

Concepts
Solved Examples
Practice Questions
Formulas

STATUS: PRODUCTION_READY (0 Critical Defects)

Important Formulas & Cheat Sheet

Variable sep: $$\int f(y)dy = \int g(x)dx$$
Integrating factor: $$\mu = e^{\int P\,dx}$$
Order = highest derivative

✍️ STEP 3: PRACTICE & ANALYZE

Differential Equations Interactive Practice

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Solve these 10 standard exam-level questions. The system tracks your response time, detects rapid guessing via Cognitive Reading Thresholds, and outputs your post-session diagnosis.

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Q1Order & Degree

Expected: 60sEasy

The temperature $T(t)$ of a body at time $t=0$ is $160^{\circ} \mathrm{F}$ and it decreases continuously as per the differential equation $\frac{d T}{d t}=-K(T-80)$ , where $K$ is a positive constant. If $T(15)=120^{\circ} \mathrm{F}$ , then $T(45)$ is equal to

Q2Variable Separable & Linear ODEs

Expected: 60sEasy

Let the population of rabbits surviving at time $t$ be governed by the differential equation ${{dp\left( t \right)} \over {dt}} = {1 \over 2}p\left( t \right) - 200.$ If $p(0)=100,$ then $p(t)$ equals:

Q3Homogeneous Equations

Expected: 60sEasy

The order and degree of the differential equation $\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}$ are

Q4Order & Degree

Expected: 60sEasy

The degree and order of the differential equation of the family of all parabolas whose axis is $x$ -axis, are respectively.

Q5Variable Separable & Linear ODEs

Expected: 90sMedium

Let y = y(x) be the solution of the differential equation, ${({x^2} + 1)^2}{{dy} \over {dx}} + 2x({x^2} + 1)y = 1$ such that y(0) = 0. If $\sqrt ay(1)$ = $\pi \over 32$ , then the value of 'a' is :

Q6Homogeneous Equations

Expected: 90sMedium

Let $y=y(t)$ be a solution of the differential equation ${{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}}$ where, $\alpha > 0,\beta > 0$ and $\gamma > 0$ . Then $\mathop {\lim }\limits_{t \to \infty } y(t)$

Q7Order & Degree

Expected: 90sMedium

Let $y = y(x)$ be the solution of the differential equation ${x^3}dy + (xy - 1)dx = 0,x > 0,y\left( {{1 \over 2}} \right) = 3 - \mathrm{e}$ . Then y (1) is equal to

Q8Variable Separable & Linear ODEs

Expected: 90sMedium

If a curve y = f(x) passes through the point (1, 2) and satisfies $x {{dy} \over {dx}} + y = b{x^4}$ , then for what value of b, $\int\limits_1^2 {f(x)dx = {{62} \over 5}}$ ?

Q9Homogeneous Equations

Expected: 120sHard

If the solution of the differential equation ${{dy} \over {dx}} + {e^x}\left( {{x^2} - 2} \right)y = \left( {{x^2} - 2x} \right)\left( {{x^2} - 2} \right){e^{2x}}$ satisfies $y(0) = 0$ , then the value of y(2) is _______________.

Q10Order & Degree

Expected: 120sHard

Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{d} y}{\mathrm{~d} x}=2 x(x+y)^3-x(x+y)-1, y(0)=1$ . Then, $\left(\frac{1}{\sqrt{2}}+y\left(\frac{1}{\sqrt{2}}\right)\right)^2$ equals :

🗺️ Recommended Learning Sequence

Integral Calculus➔Differential Equations (Current)➔Coordinate Geometry

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Differential Equations Preparation Hub

What is Differential Equations?

💡 Why Study Differential Equations?

⚠️ Common Pitfalls to Avoid

Differential Equations Theory & Derivations

Differential Equations

Overview

Subtopics

Important Formulas & Cheat Sheet

Differential Equations Interactive Practice

Let the population of rabbits surviving at time ttt be governed by the differential equation dp(t)dt=12p(t)−200.{{dp\left( t \right)} \over {dt}} = {1 \over 2}p\left( t \right) - 200.dtdp(t)​=21​p(t)−200. If p(0)=100,p(0)=100,p(0)=100, then p(t)p(t)p(t) equals:

The order and degree of the differential equation (1+3dydx)2/3=4d3ydx3\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}(1+3dxdy​)2/3=4dx3d3y​ are

The degree and order of the differential equation of the family of all parabolas whose axis is xxx-axis, are respectively.

Let y = y(x) be the solution of the differential equation, (x2+1)2dydx+2x(x2+1)y=1{({x^2} + 1)^2}{{dy} \over {dx}} + 2x({x^2} + 1)y = 1(x2+1)2dxdy​+2x(x2+1)y=1 such that y(0) = 0. If ay(1)\sqrt ay(1)a​y(1) = π32\pi \over 3232π​ , then the value of 'a' is :

Let y=y(x)y = y(x)y=y(x) be the solution of the differential equation x3dy+(xy−1)dx=0,x>0,y(12)=3−e{x^3}dy + (xy - 1)dx = 0,x > 0,y\left( {{1 \over 2}} \right) = 3 - \mathrm{e}x3dy+(xy−1)dx=0,x>0,y(21​)=3−e. Then y (1) is equal to

If a curve y = f(x) passes through the point (1, 2) and satisfies xdydx+y=bx4x {{dy} \over {dx}} + y = b{x^4}xdxdy​+y=bx4, then for what value of b, ∫12f(x)dx=625\int\limits_1^2 {f(x)dx = {{62} \over 5}} 1∫2​f(x)dx=562​?

🗺️ Recommended Learning Sequence

Ready to achieve complete mastery in Differential Equations?

Let the population of rabbits surviving at time $t$ be governed by the differential equation ${{dp\left( t \right)} \over {dt}} = {1 \over 2}p\left( t \right) - 200.$ If $p(0)=100,$ then $p(t)$ equals:

The order and degree of the differential equation $\,{\left( {1 + 3{{dy} \over {dx}}} \right)^{2/3}} = 4{{{d^3}y} \over {d{x^3}}}$ are

The degree and order of the differential equation of the family of all parabolas whose axis is $x$ -axis, are respectively.

Let y = y(x) be the solution of the differential equation, ${({x^2} + 1)^2}{{dy} \over {dx}} + 2x({x^2} + 1)y = 1$ such that y(0) = 0. If $\sqrt ay(1)$ = $\pi \over 32$ , then the value of 'a' is :

Let $y = y(x)$ be the solution of the differential equation ${x^3}dy + (xy - 1)dx = 0,x > 0,y\left( {{1 \over 2}} \right) = 3 - \mathrm{e}$ . Then y (1) is equal to

If a curve y = f(x) passes through the point (1, 2) and satisfies $x {{dy} \over {dx}} + y = b{x^4}$ , then for what value of b, $\int\limits_1^2 {f(x)dx = {{62} \over 5}}$ ?