These notes for CBSE Class 12 Mathematics, Chapter 9, focus on Differential Equations. They define a differential equation as an equation involving variables and their derivatives. The notes explain how to determine the order and degree of a differential equation, distinguishing between linear and non-linear types. A key section covers the solutions of differential equations, including general and particular solutions, and how they are also referred to as primitives. The process of forming a differential equation from a given relation by eliminating arbitrary constants is detailed. Various methods for solving first-order, first-degree differential equations are presented, such as the inspection method, variable separable method, homogeneous equations, and reducible forms. Additionally, the notes touch upon exact differential equations, linear differential equations, Bernoulli's equation, Clairaut's form, and the concept of orthogonal trajectories. These notes are designed to aid students in revising the core concepts of differential equations for their exams.
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An equation that involves an independent variable, dependent variable and differential coefficients of dependent variable with respect to the independent variable is called a differential equation. e.g., (i) x2(d2y / dx2) + x3 (dy / dx)3 7x2y2 (ii) (x2 + y2) dx = (x2 – y2) dy
The order of a differential equation is the order of the highest derivative occurring in the
equation. The order of a differential equation is always a positive integer.
The degree of a differential equation is the degree (exponent) of the derivative of the highest
order in the equation, after the equation is free from negative and fractional powers of the
derivatives.
A differential equation is said to be linear, if the dependent variable and all of its derivatives
occurring in the first power and there are no product of these. A linear equation of nth order
can be written in the form
where, P0, P1, P2,…, Pn – 1 and Q must be either constants or functions of x only.
A linear differential equation is always of the first degree but every differential equation of the
first degree need not be linear.
e.g., The equations d2y / dx2 + (dy / dx)2 + xy = 0 and
x(d2y / dx2) + y (dy / dx) + y = x3, (dy / dx) d2y / dx2 + y = 0
are not linear.
A solution of a differential equation is a relation between the variables, not involving the
differential coefficients, such that this relation and the derivative obtained from it satisfy the
given differential equation.
e.g., Let d2y / dx2 + y = 0
Integrating above equation twicely, we get y = A cos x + B sin x.
If the solution of the differential equation contains as many independent arbitrary constants as
the order of the differential equation, then it is called the general solution or the complete
integral of the differential equation.
e.g., The general solution of d2y / dx2 + y = 0 is y = A cos x + B sin x because it contains two
arbitrary constants A and B, which is equal to the order of the equation.
Solution obtained by giving particular values to the arbitrary constants in the general solution is called a particular solution. e.g., In the previous example, if A = B = 1, then y = cos x + sin x is a particular solution of the differential equation d2y / dx2 + y = 0.
Solution of a differential equation is also called its primitive.
Suppose, we have a given equation with n arbitrary constants f(x, y, c1, c2,…, cn) = 0. Differentiate the equation successively n times to get n equations. Eliminating the arbitrary constants from these n + 1 equations leads to the required differential equations.
A differential equation of first degree and first order can be solved by following method.
If the differential equation’ can be written as f [f1(x, y) d {f1(x, y)}] + ? [f2(x, y) d {f2(x, y)}]
+… = 0] then each term can be integrated separately.
For this, remember the following results
If the equation can be reduced into the form f(x) dx + g(y) dy = 0, we say that the variable have been separated. On integrating this reduced, form, we get ? f(x) dx + ? g(y) dy = C, = C, where C is any arbitrary constant.
A differential equation of the form dy / dx = f(ax + by + c) can be reduced to variables separable form by substituting ax + by + c = z => a + b dy / dx = dz / dx The given equation becomes 1 / b (dz / dx – a) f(z) => dz / dx = a + b f(z) => dz / a+ bf(z) = dx Hence, the variables are separated in terms of z and x.
A function f(x, y) is said to be homogeneous of degree n, if f(?x, ?y) = ?n f(x, y) Suppose a differential equation can be expressed in the form dy / dx = f(x, y) / g(x, y) = F (y / x) where, f(x, y) and g(x, y) are homogeneous function of same degree. To solve such types of equations, we put y = vx => dy / dx = v + x dv / dx. The given equation, reduces to v + x dv / dx = F(v) => x dv / dx = F(v) – v ? dv / F(v) – v = dx / x Hence, the variables are separated in terms of v and x.
The differential equation of the form dy / dx = a1x + b1y + c1 / a2x + b2y + c2 …… (i) put X = X + h and y = Y + k
? dY / dX = a1 X + b1 Y + (a1h + b1k + c1) / a2X + b2 Y + (a2h + b2k + c2) ……(ii)
We choose h and k, so as to satisfy a1h + b1k + c1 = 0 and a2h + b2k + c2 = 0.
On solving, we get
h / b1c2 – b2c1 = k / c1a2 – c2a1 = 1 / a1b2 – a2b1
? h = b1c2 – b2c1 / a1b2 – a2b1 and k = c1a2 – c2a1 / a1b2 – a2b1
Provided a1b2 – a2b1 ? 0 , a1 / a2 ? ba / b2
Then, Eq, (ii) reduces to dY / dX = (a1 X + b1 Y) / (a2X + b2 Y), which is a homogeneous form
and will be solved easily.
Differential equation M(x,y) dy + N(x,y) dy = 0 is called an exact differential equation. If a function u (x, y) exist such that, du = Mdx + Ndy.
Differential equation Mdx + Ndy = 0 where, M and N are the functions •of x and y, will be an exact differential equation, if?N / ?y = ?N / ?x
A linear differential equation of the first order can be either of the following forms
(i) dy / dx + Py = Q, where P and Q are functions of x or constants.
(ii) dx / dy + Rx = S, where Rand S are functions of y or constants.
Consider the differential Eq. (i) i.e., dy / dx + Py = Q
Similarly, for the second differential equation dx / dy + Rx = S, the integrating factor, IF = e ?R
dy and the general solution is
x (IF) = ? S (IF) dy + C
Bernoulli’s Equation An equation of the form dy / dx + Py = Qyn, where P and Q are functions of x along or constants, is called Bernoulli’s equation. Divide both the sides by yn, we get y-n dy / dx + Py-n + 1 = Q Put y-n + 1 = z => (-n + 1)y-n dy / dx = dz / dx The equation reduces to 1 / 1 – n dz / dx + Pz = Q => dz / dx + (1 – n) Pz = Q (1 – n) which is linear in z and can be solved in the usual manner.
Differential equation y = Px + f(p), where P= dy / dx … (i) is called clairaut form of differential equation. In which, get its general solution by replacing P from C. Now, differential on both sides of Eq, (i) with respect to x and put dy / dx = P. P = P + x dp / dx + f’ (P) dp / dx = 0 => [x + f’ (p)] dp / dx = 0 => dp / dx = 0 => p = C
Any curve, which cuts every member of a given family of curves at right angle, is called anorthogonal trajectory of the family.
(i) Let f(x,y,c)= 0 be the equation of the given family of curves, where ‘c’ is an arbitrary
parameter.
(ii) Differentiate f = 0, with respect to ‘x’ and eliminate 0, i.e., from a differential equation.
(iii) Substitute (- dx / dy) for (dy / dx) in the above differential equation. This will give the differential equation of the orthogonal trajectories.
(iv) By solving this differential equation, we get the required orthogonal trajectories.
A differential equation is an equation that involves an independent variable, a dependent variable, and differential coefficients of the dependent variable with respect to the independent variable.
The order of a differential equation is the order of the highest derivative present in the equation. It is always a positive integer.
The degree of a differential equation is the exponent of the highest order derivative after the equation is free from negative and fractional powers of derivatives.
A general solution contains as many independent arbitrary constants as the order of the differential equation. A particular solution is obtained by assigning specific values to these arbitrary constants in the general solution.
This method is used when a differential equation can be reduced to the form f(x) dx + g(y) dy = 0. Integrating both sides gives the solution.
A differential equation is homogeneous if it can be expressed in the form dy/dx = F(y/x), where F is a function of y/x. It is solved by substituting y = vx.
An orthogonal trajectory is a curve that intersects every member of a given family of curves at a right angle.
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