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If you are studying Degree / BSc 1st Year 1st Sem Maths, then Differential integral calculus is one of the most important and scoring papers. Many students feel this subject is difficult because they are not sure what topics to study and which questions are important for exams.
This article provides:
Unit-wise syllabus topics
Unit-wise important questions
Exam-focused preparation guidance
The content is written in simple, student-friendly language, especially useful for average and weak students.
Paper Name: Differential integral calculus
Course: Degree / BSc Mathematics
Year: 1st Year
Semester: 1st Semester
Total Units: Unit 1 to Unit 4
Unit 1: Differential Equations of First Order & First Degree
Unit 2: Differential Equations of First Order but Not of First Degree and Applications
Unit 3: Higher Order Linear Differential Equations
Unit 4: Higher Order Linear Differential Equations and Partial Differential Equations
Before solving problems, students must understand these topics:
Introduction – Equations in which variables are separable
Homogeneous differential equations
Differential equations reducible to homogeneous form
Linear differential equations
Differential equations reducible to linear form
Exact differential equations
Integrating factors
Change of variables
Total differential equations
Simultaneous total differential equations of the form
dxP=dyQ=dzR\frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}Pdx=Qdy=Rdz
👉 These topics form the foundation of Differential Equations.
1. Solve: \(\frac{dy}{dx} + y \tan x = \sin x\)
2. Solve: \((x^2 + y^2)dx - 2xy \, dy = 0\)
3. Solve: \((2xy + y^2)dx + (x^2 + 2xy)dy = 0\)
4. Solve: \(\frac{dy}{dx} + y^2 = xy\)
5. Solve (by change of variables): \((x^2 + y^2)dx - 2xy \, dy = 0\)
6. Solve: \((3x^2 + 2y)dx + (2x - y^2)dy = 0\)
7. Solve: \(\frac{dy}{dx} = \frac{x - y}{x + y}\)
8. Solve: \(\frac{dy}{dx} + \frac{x}{y} = \sin x\)
9. Find the integrating factor and solve: \((y + 2x)dx + (x + 3y)dy = 0\)
10. Solve: \(x \, dy - y \, dx = (x^2 + y^2)dx\)
Equations solvable for p
Equations solvable for x
Equations not containing x or y
Homogeneous in x and y – Clairaut’s equation
First degree in x and y differential equations
Growth and decay – dynamics of tumour growth
Radioactivity and carbon dating
Compound interest
Orthogonal trajectories
1. Solve: \(y = px + a^2 + p^2\)
2. Solve Clairaut’s equation: \(y = px + f(p)\)
3. Solve: \(\frac{dy}{dx} = \frac{x - y}{x + y}\)
4. Solve: \(y^2 = 2px + p^2\)
5. Solve: \(x^2 = 2py + p^2\)
6. Solve: \(p^2 + y^2 = 1\)
7. Solve: \(\frac{dy}{dx} = \frac{2xy}{x^2 + y^2}\)
8. Application: Solve the differential equation for compound interest.
9. Application: Solve the differential equation for radioactive decay.
10. Application: Find the orthogonal trajectories of \(x^2 + y^2 = c^2\)
Solution of homogeneous linear differential equations with constant coefficients
Solution of non-homogeneous differential equations
\(P(D)y = Q(x)\)
Method of Undetermined Coefficients
1. When \(Q(x) = V e^{ax}\)
2. When \(Q(x) = b \sin(ax)\) or \(Q(x) = b \cos(ax)\)
3. When \(Q(x) = b x^{k}\)
4. When \(Q(x) = b e^{ax}\)
1. Solve: \((D^2 - 2D + 1)y = e^x\)
2. Solve: \((D^2 + 9)y = \sin 3x\)
3. Solve: \((D^2 + 4)y = x^2\)
4. Solve: \((D^2 - 1)y = \cosh x\)
5. Solve: \((D^3 - 3D^2 + 3D - 1)y = e^x\)
6. Solve: \((D^2 + 2D + 1)y = e^{-x}\)
7. Solve: \((D^2 - 4)y = \sin 2x\)
8. Solve: \((D^2 + 1)y = x \cos x\)
9. Solve: \((D^3 - D)y = x^2\)
10. Solve: \((D^2 + 9)y = x^2 + \cos 3x\)
Method of variation of parameters
Linear differential equations with non-constant coefficients
Cauchy–Euler equation
Legendre’s linear equations
Miscellaneous differential equations
Partial differential equations:
Formation of partial differential equations
Solution of partial differential equations
Equations easily integrable
Linear equations of first order
1. Solve by variation of parameters: \((D^2 + 1)y = \tan x\)
2. Solve: \(x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} - y = 0\)
3. Solve: \((1 - x^2)\frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + n(n+1)y = 0\)
4. Solve by variation of parameters: \((D^2 + 4)y = \sec 2x\)
5. Solve: \(x^2 \frac{d^2y}{dx^2} + 3x \frac{dy}{dx} + y = 0\)
6. Solve: \((1 - x^2)\frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0\)
7. Solve total differential equation: \((y+z)dx + (z+x)dy + (x+y)dz = 0\)
8. Solve simultaneous differential equations: \(\frac{dx}{y+z} = \frac{dy}{z+x} = \frac{dz}{x+y}\)
9. Solve by variation of parameters: \((D^2 + 1)y = \frac{\sin x}{x}\)
10. Solve: \(x^2 \frac{d^2y}{dx^2} - x \frac{dy}{dx} + y = 0\)
Follow this simple method:
Read the full syllabus once
Study important questions unit-wise
Practice writing answers
Revise important questions again before exams
👉 This method is very effective for 1st year students.
Many students:
Study everything without focus
Skip writing practice
Start preparation very late
Panic before exams
👉 Studying important questions early avoids these mistakes.
Important questions help in:
Passing exams
Scoring decent marks
But for very high marks, students should also:
Understand concepts
Practice textbook problems
👉 Important questions are a smart starting point, not shortcut cheating.
Yes, in most cases, important questions are enough to pass if studied properly.
Yes. Many average students pass and score well using unit-wise important questions.
Yes. The content is useful for both Degree and BSc 1st year students.
Yes. Similar questions are repeated every year with small changes.
Differential Equations may look difficult at first, but with proper planning and the right set of important questions, it becomes much easier to handle. This article is designed to help Degree / BSc 1st Year 1st Semester students focus on exam-oriented preparation without confusion.
If you study the unit-wise topics, practice the important questions regularly, and revise them before exams, you can confidently pass and score well. Remember, understanding the method is more important than memorizing steps.
Use this page as your main revision guide for Differential Equations. Start early, practice neatly, and stay calm during exams — success will follow.
“Built for clarity, not cramming”
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