Mathematicians including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d’Alembert, and Euler have studied differential equations and contributed to the field. Hence we can estimate the importance and popularity of the this branch of mathematics.
A simple example is Newton’s second law of motion — the relationship between the displacement x and the time t of an object under the force F, is given by the differential equation,
which constrains the motion of a particle of constant mass m. This equation is probably the most popular Ordinary Differential Equation.
Definition 1.1– A differential equation involving derivatives with respect to a single independent variable is called an ordinary differential equation.
Linear and non-linear differential equations–
Definition 1.2– A differential equation is called linear if (i) every dependent variable and every derivative involved occurs in the first degree only, and (ii) no products of dependent variables and/or derivatives occur. A differential equation which is not linear is called a non-linear differential equation.
Solution of a differential equation–
Definition 1.3– Any relation between the dependent and independent variables, when substituted in the differential equation, reduces it to an identity is called a solution or integral of the differential equation. It should be noted that a solution of a differential equation does not involve the derivatives of the dependent variable with respect to the independent variable or variables. For example, y = ce^2x is a solution of dy/dx = 2y because by putting y = ce^2x and dy/dx = 2ce2x, the given differential equation reduces to the identity 2ce^2x = 2ce^2x. Observe that y = ce^2x is a solution of the given differential equation for any real constant c which is called an arbitrary constant.
Family of curves–
Definition 1.4– An n-parameter family of curves is a set of relations of the form {(x, y) : f (x, y, c1, c2, …, cn) = 0}, where ‘f ’ is a real valued function of x, y, c1, c2, …, cn and each ci (i = 1, 2, …, n) ranges over an interval of real values. For example, the set of concentric circles defined by x2 + y2 = c is one parameter family if c takes all non-negative real values. Again, the set of circles, defined by (x – c1)^2 + (y – c2)^2 = c3 is a three-parameter family if c1, c2 take all real values and c3 takes all non-negative real values.
TYPES OF SOLUTIONS
Let F (x, y, y1, y2, …, yn) = 0 ……….. (1) be an nth order ordinary differential equation.
- Complete primitive or, General solution A solution of (1) containing n independent arbitrary constants is called a general solution.
- Particular solution A solution of (1) obtained from a general solution of (1) by giving particular values to one or more of the n independent arbitrary constants is called a particular solution of (1).
- Singular solution A solution of (1) which cannot be obtained from any general solution of (1) by any choice of the n independent arbitrary constants is called a singular solution of (1).
Working rule to form the differential equation from the given equation in x and y, containing n arbitrary constants.
Step I. Write the equation of the given family of curves.
Step II. Differentiate the equation of step I, n times so as to get n additional equations containing the n arbitrary constants and derivatives.
Step III. Eliminate n arbitrary constants from the (n + 1) equations obtained in steps I and II. Thus, we obtain the required differential equation involving a derivative of nth order.
Solved example–
Q. Find the differential equation of the family of curves y = e^mx, where m is an arbitrary constant.
Sol. Given that y = e^mx. …… (1)
Differentiating (1) w.r.t. ‘x’, we get dy/dx = me^mx. …… (2)
Now, (1) and (2) , dy/dx = my , m = (1/y) × (dy/dx). …… (3)
Again, from (1), mx = ln y so that m = (ln y)/x. …… (4)
Eliminating m from (3) and (4), we get (1/y) × (dy/dx) = (1/x) × ln y.
Q. (a) Find the differential equation of all straight lines passing through the origin. (b) Find the differential equation of all the straight lines in the xy-plane.
Sol. (a) Equation of any straight line passing through the origin is
y = mx, m being arbitrary constant. …… (1)
Differentiating (1) w.r.t. ‘x’, dy/dx = m. …… (2)
Eliminating m from (1) and (2), we get y = x (dy/dx).
(b) We know that equation of any straight line in the xy-plane is given by
y = mx + c, m and c being arbitrary constants. …… (1)
Differentiating (1) w.r.t. ‘x’, we get dy/dx = m. …… (2)
Differentiating (2) w.r.t. ‘x’, we get d2y/dx2 = 0, …… (3)
which is the required differential equation.
MATLAB PLOT FOR ODE SOLUTION
SOURCE CODE
w = 1;
k=1;
figure
tspan = linspace(0, 5); % Create Constant ‘tspan’
zv=0.1:0.01:0.5; % Vector Of ‘z’ Values
gs2 = zeros(numel(tspan), numel(zv)); % Preallocate
for k = 1:numel(zv)
z = zv(k);
f = @(t,x) [-1i.*(2*w + 2*z).*x(1) + -1i.*sqrt(2).*k.*x(2);-1i.*sqrt(2).*k.*x(1) + -1i.*2*w*x(2)+-1i.*sqrt(2).*k.*x(3);-1i.*sqrt(2).*k.*x(2)+-1i.*2*w*x(3)];
[t,xa] = ode45(f,tspan,[0 1 0]);
gs = abs(xa).^2;
gs2(:,k) = gs(:,2); % Save Second Column Of ‘gs’ In ‘gs2’ Matrix
end
figure
surf(t,zv,gs2')
grid on
xlabel('t')
ylabel('z')
shading('interp')Linearly dependent and independent set of functions
Definition. n functions y1 (x), y2 (x), …, yn (x) are linearly dependent if there exist constants c1, c2, …, cn (not all zero), such that c1 y1 + c2 y2 + … + cn yn = 0.
If, however, the identity implies that c1 = c2 = … = cn = 0, then y1, y2, …, yn are said to be linearly independent.
Existence and uniqueness theorem
Consider a second order linear differential equation of the form a0 (x) y” + a1 (x) y’ + a2 (x) y = r (x), … (1) where a0 (x), a1 (x), a2 (x) and r (x) are continuous functions on an interval (a, b) and a0 (x) ; 0 for each x in (a, b). Let c1 and c2 be arbitrary real numbers and x0 in (a, b). Then there exists a unique solution y (x) of (1) satisfying y (x0) = c1 and y’ (x0) = c2. Moreover, this solution y (x) is defined over the interval (a, b).