The solution of a differential equation – General and particular will use integration in some steps to solve it. We will be learning how to solve a differential equation with the help of solved examples. Also learn to the general solution for first-order and second-order differential equation.
Table of Contents
Let us first understand to solve a simple case here:
Consider the following equation: 2x2 – 5x – 7 = 0. The solution to this equation is a number i.e. -1 or 7/2 which satisfies the above equation. Which means putting the value of variable x as -1 or 7/2, we get Left-hand side (LHS) equal to Right-hand side (RHS) i.e 0. But in the case of the differential equation, the solution is a function that satisfies the given differential equation. That means we need to differentiate the given equation first and then find the solutions for it. The differential equations examined are of the form y’ = /(x, y) (equations of higher orders could be reduced to equations of the first order). The function f is considered to be analytic in a adequately large neighbourhood of the initial point (x0,y0).
General Solution of a Differential Equation
When the arbitrary constant of the general solution takes some unique value, then the solution becomes the particular solution of the equation.
By using the boundary conditions (also known as the initial conditions) the particular solution of a differential equation is obtained.
So, to obtain a particular solution, first of all, a general solution is found out and then, by using the given conditions the particular solution is generated.
Suppose,
dy/dx = ex + cos2x + 2x3
Then we know, the general solution is:
y = ex + sin2x/2 + x4/2 + C
Now, x = 0, y = 5 substituting this value in the general solution we get,
5 = e0 + sin(0)/2 + (0)4/2 + C
C = 4
Hence, substituting the value of C in the general solution we obtain,
y = ex + sin2x/2 + x4/2 + 4
This represents the particular solution of the given equation.
General Solution for First Order and Second Order
If we have to solve a first-order differential equation by variable separable method, we need have to mention an arbitrary constant before we start performing integration. Hence, we can see that a solution of the first-order differential equation has at least one fixed arbitrary constant after simplification.
Variable separable differential Equations: The differential equations which are represented in terms of (x,y) such as the x-terms and y-terms can be ordered to different sides of the equation (including delta terms). Thus, each variable after separation can be integrated easily to find the solution of the differential equation. The equations can be written as:
f(x)dx+g(y)dy=0, where f(x) and g(y) are either constants or functions of x and y respectively.
Similarly, the general solution of a second-order differential equation will consist of two fixed arbitrary constants and so on. The general solution geometrically interprets an m-parameter group of curves.
Particular Solution of a Differential Equation
A Particular Solution is a solution of a differential equation taken from the General Solution by allocating specific values to the random constants. The requirements for determining the values of the random constants can be presented to us in the form of an Initial-Value Problem, or Boundary Conditions, depending on the query.
Singular Solution
The Singular Solution of a given differential equation is also a type of Particular Solution but it can’t be taken from the General Solution by designating the values of the random constants.
Differential Equations Example
Example: dy/dx = x2
Solution: dy = x2 dx
Integrating both sides, we get
\(\begin{array}{l}\Rightarrow \int dy = \int x^2 dx \end{array} \)
If we solve this equation to figure out the value of y we get
y = x3/3 + C
where C is an arbitrary constant.
In the above-obtained solution, we see that y is a function of x. On substituting this value of y in the given differential equation, both the sides of the differential equation becomes equal.
Differential Equations Practice Problems with Solutions
The solution obtained above after integration consists of a function and an arbitrary constant. This represents a general solution of the given equation.
Let the solution be represented as:
\(\begin{array}{l} y = \phi(x) + C \end{array} \)
It represents the solution curve or the integral curve of the given differential equation.
Thus, we can say that a general solution always involves a constant C.
Let us consider some more examples:
Solved Examples
To get a better insight into the topic, let us have a look at the following example.
Example – Find out the particular solution of the differential equation ln dy/dx = e4y + ln x, given that for x = 0, y = 0.
Solution –dy/dx = e4y + ln x
dy/dx = e4y × eln x
dy/dx = e4y × x
1/e4ydy = x dx
e-4ydy = x dx
Integrating both the sides with respect to y and x respectively we get,
e−4y/−4=x2/2+C
This represents the general solution of the differential equation given.
Now, it is also given that y(0)=0, substituting this value in the above general solution we get,
e0/−4=02/2+C
⇒C=−1/4
Hence, the above equation can be rewritten as
e−4y/−4=x2/2–14
⇒e−4y=−2x2+1
⇒ln(e−4y)=ln(1−2x2)
⇒−4y=ln(1−2x2)
⇒y=–ln(1−2x2)/4
which is the particular solution of the differential equation given.
Neeraj Anand, Param Anand
Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations.
In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS".
He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.
CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution
The table below shows the marks weightage along with the number of periods required for teaching. The Maths theory paper is of 80 marks, and the internal assessment is of 20 marks which totally comes out to be 100 marks.
CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24
Max Marks: 80
No.
Units
Marks
I.
Relations and Functions
08
II.
Algebra
10
III.
Calculus
35
IV.
Vectors and Three – Dimensional Geometry
14
V.
Linear Programming
05
VI.
Probability
08
Total Theory
80
Internal Assessment
20
Grand Total
100
Unit-I: Relations and Functions
1. Relations and Functions
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.
2. Inverse Trigonometric Functions
Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.
Unit-II: Algebra
1. Matrices
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants
Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.
Unit-III: Calculus
1. Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions like sin-1 x, cos-1 x and tan-1 x, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.
2. Applications of Derivatives
Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
3. Integrals
Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals
Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)
5. Differential Equations
Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:
dy/dx + py = q, where p and q are functions of x or constants.
dx/dy + px = q, where p and q are functions of y or constants.
Unit-IV: Vectors and Three-Dimensional Geometry
1. Vectors
Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.
2. Three – dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.
Unit-V: Linear Programming
1. Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Unit-VI: Probability
1. Probability
Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.
Students can go through the CBSE Class 12 Syllabus to get the detailed syllabus of all subjects. Get access to interactive lessons and videos related to Maths and Science with ANAND CLASSES’S App/ Tablet.
Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26
Q1
Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?
Yes, Calculus is an important chapter in the CBSE Class 12 Maths Syllabus. It is for 35 marks which means that if a student is thorough with this chapter will be able to pass the final exam.
Q2
How many units are discussed in the CBSE Class 12 Maths Syllabus?
In the CBSE Class 12 Maths Syllabus, about 6 units are discussed, which contains a total of 13 chapters.
Q3
How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?
About 20 marks are allotted for internals in the CBSE Class 12 Maths Syllabus. Students can score it with ease through constant practice.
Anand Technical Publishers
Buy Products (Printed Books & eBooks) of Anand Classes published by Anand Technical Publishers, Visit at following link :
