{"id":354,"date":"2020-08-05T06:10:40","date_gmt":"2020-08-05T06:10:40","guid":{"rendered":"http:\/\/soulofmathematics.com\/?page_id=354"},"modified":"2020-09-02T10:40:55","modified_gmt":"2020-09-02T05:10:55","slug":"ordinary-differential-equation","status":"publish","type":"page","link":"https:\/\/soulofmathematics.com\/index.php\/ordinary-differential-equation\/","title":{"rendered":"ORDINARY DIFFERENTIAL EQUATION"},"content":{"rendered":"\n

Mathematicians including Newton<\/a>, Leibniz<\/a>, the Bernoulli family<\/a>, Riccati<\/a>, Clairaut<\/a>, d’Alembert<\/a>, and Euler<\/a> have studied differential equations and contributed to the field. Hence we can estimate the importance and popularity of the this branch of mathematics.<\/p>\n\n\n\n

A simple example is Newton’s second law<\/a> of motion \u2014 the relationship between the displacement x<\/em> and the time t<\/em> of an object under the force F<\/em>, is given by the differential equation,<\/p>\n\n\n\n

\"{\\displaystyle<\/figure>\n\n\n\n

which constrains the motion of a particle of constant mass m<\/em>. This equation is probably the most popular Ordinary Differential Equation.<\/p>\n\n\n\n

Definition<\/strong> 1.1<\/strong>– A differential equation involving derivatives with respect to a single independent variable is called an ordinary differential equation.<\/p>\n\n\n\n

Linear and non-linear differential equations<\/strong>–<\/p>\n\n\n\n

Definition 1.2<\/strong>– 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.<\/p>\n\n\n\n

Solution of a differential equation<\/strong>–<\/p>\n\n\n\n

Definition 1.3<\/strong>– 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.<\/p>\n\n\n\n

Family of curves<\/strong>–<\/p>\n\n\n\n

Definition 1.4<\/strong>– An n-parameter family of curves is a set of relations of the form {(x, y) : f (x, y, c1, c2, \u2026, cn) = 0}, where \u2018f \u2019 is a real valued function of x, y, c1, c2, \u2026, cn and each ci (i = 1, 2, \u2026, 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 \u2013 c1)^2 + (y \u2013 c2)^2 = c3 is a three-parameter family if c1, c2 take all real values and c3 takes all non-negative real values.<\/p>\n\n\n\n

TYPES OF SOLUTIONS<\/h5>\n\n\n\n

Let F (x, y, y1, y2, \u2026, yn) = 0 \u2026…….. (1) be an nth order ordinary differential equation.<\/p>\n\n\n\n

  1. Complete primitive or, General solution<\/strong> A solution of (1) containing n independent arbitrary constants is called a general solution.<\/li>
  2. Particular solution<\/strong> 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).<\/li>
  3. Singular solution<\/strong> 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). <\/li><\/ol>\n\n\n\n

    Working rule to form the differential equation from the given equation in x and y, containing n arbitrary constants.<\/strong>
    Step I<\/strong>. Write the equation of the given family of curves.
    Step II<\/strong>. Differentiate the equation of step I, n times so as to get n additional equations containing the n arbitrary constants and derivatives.
    Step III<\/strong>. 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.<\/p>\n\n\n\n

    Solved example<\/strong>–<\/p>\n\n\n\n

    Q<\/strong>. Find the differential equation of the family of curves y = e^mx, where m is an arbitrary constant.<\/em><\/p>\n\n\n\n

    Sol<\/strong>. Given that y = e^mx. \u2026… (1)
    Differentiating (1) w.r.t. \u2018x\u2019, we get dy\/dx = me^mx. \u2026… (2)
    Now, (1) and (2) , dy\/dx = my , m = (1\/y) \u00d7 (dy\/dx). \u2026… (3)
    Again, from (1), mx = ln y so that m = (ln y)\/x. \u2026… (4)
    Eliminating m from (3) and (4), we get (1\/y) \u00d7 (dy\/dx) = (1\/x) \u00d7 ln y.<\/p>\n\n\n\n

    Q<\/strong>. (a)<\/strong> Find the differential equation of all straight lines passing through the origin. (b)<\/strong> Find the differential equation of all the straight lines in the xy-plane.<\/em><\/p>\n\n\n\n

    Sol<\/strong>. (a)<\/strong> Equation of any straight line passing through the origin is
    y = mx, m being arbitrary constant. \u2026… (1)
    Differentiating (1) w.r.t. \u2018x\u2019, dy\/dx = m. \u2026… (2)
    Eliminating m from (1) and (2), we get y = x (dy\/dx).
    (b)<\/strong> We know that equation of any straight line in the xy-plane is given by
    y = mx + c, m and c being arbitrary constants. \u2026… (1)
    Differentiating (1) w.r.t. \u2018x\u2019, we get dy\/dx = m. \u2026… (2)
    Differentiating (2) w.r.t. \u2018x\u2019, we get d2y\/dx2 = 0, \u2026… (3)
    which is the required differential equation.<\/p>\n\n\n\n

    MATLAB PLOT FOR ODE SOLUTION<\/strong><\/p>\n\n\n\n

    \"Matlab<\/figure>\n\n\n\n

    SOURCE CODE<\/p>\n\n\n\n

    w = 1;\nk=1;\nfigure\ntspan = linspace(0, 5); % Create Constant \u2018tspan\u2019\nzv=0.1:0.01:0.5; % Vector Of \u2018z\u2019 Values\ngs2 = zeros(numel(tspan), numel(zv)); % Preallocate\nfor k = 1:numel(zv)\nz = zv(k);\nf = @(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)];\n[t,xa] = ode45(f,tspan,[0 1 0]);\ngs = abs(xa).^2;\ngs2(:,k) = gs(:,2); % Save Second Column Of \u2018gs\u2019 In \u2018gs2\u2019 Matrix\nend\n\nfigure\nsurf(t,zv,gs2')\ngrid on\nxlabel('t')\nylabel('z')\nshading('interp')<\/pre>\n\n\n\n
    Linearly dependent and independent set of functions<\/strong><\/p>\n\n\n\n
    Definition.<\/strong> n functions y1 (x), y2 (x), \u2026, yn (x) are linearly dependent if there exist constants c1, c2, \u2026, cn (not all zero), such that c1 y1 + c2 y2 + \u2026 + cn yn = 0.<\/p>\n\n\n\n
    If, however, the identity implies that c1 = c2 = \u2026 = cn = 0, then y1, y2, \u2026, yn are said to be linearly independent.<\/p>\n\n\n\n
    Existence and uniqueness theorem<\/strong><\/p>\n\n\n\n
    Consider a second order linear differential equation of the form a0 (x) y” + a1 (x) y’ + a2 (x) y = r (x), \u2026 (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).<\/p>\n\n\n\n
    <\/p>\n\n\n\n
    <\/p>\n\n\n