{"version":"1.0","provider_name":"SOUL OF MATHEMATICS","provider_url":"https:\/\/soulofmathematics.com","author_name":"Rajarshi Dey","author_url":"https:\/\/soulofmathematics.com\/index.php\/author\/rajarshidey1729gmail-com\/","title":"ORDINARY DIFFERENTIAL EQUATION - SOUL OF MATHEMATICS","type":"rich","width":600,"height":338,"html":"<blockquote class=\"wp-embedded-content\" data-secret=\"ZpsuvHWwcS\"><a href=\"https:\/\/soulofmathematics.com\/index.php\/ordinary-differential-equation\/\">ORDINARY DIFFERENTIAL EQUATION<\/a><\/blockquote><iframe sandbox=\"allow-scripts\" security=\"restricted\" src=\"https:\/\/soulofmathematics.com\/index.php\/ordinary-differential-equation\/embed\/#?secret=ZpsuvHWwcS\" width=\"600\" height=\"338\" title=\"&#8220;ORDINARY DIFFERENTIAL EQUATION&#8221; &#8212; SOUL OF MATHEMATICS\" data-secret=\"ZpsuvHWwcS\" frameborder=\"0\" marginwidth=\"0\" marginheight=\"0\" scrolling=\"no\" class=\"wp-embedded-content\"><\/iframe><script type=\"text\/javascript\">\n\/* <![CDATA[ *\/\n\/*! This file is auto-generated *\/\n!function(d,l){\"use strict\";l.querySelector&&d.addEventListener&&\"undefined\"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!\/[^a-zA-Z0-9]\/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret=\"'+t.secret+'\"]'),o=l.querySelectorAll('blockquote[data-secret=\"'+t.secret+'\"]'),c=new RegExp(\"^https?:$\",\"i\"),i=0;i<o.length;i++)o[i].style.display=\"none\";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute(\"style\"),\"height\"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):\"link\"===t.message&&(r=new URL(s.getAttribute(\"src\")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener(\"message\",d.wp.receiveEmbedMessage,!1),l.addEventListener(\"DOMContentLoaded\",function(){for(var e,t,s=l.querySelectorAll(\"iframe.wp-embedded-content\"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute(\"data-secret\"))||(t=Math.random().toString(36).substring(2,12),e.src+=\"#?secret=\"+t,e.setAttribute(\"data-secret\",t)),e.contentWindow.postMessage({message:\"ready\",secret:t},\"*\")},!1)))}(window,document);\n\/* ]]> *\/\n<\/script>\n","thumbnail_url":"https:\/\/i0.wp.com\/soulofmathematics.com\/wp-content\/uploads\/2020\/08\/5xNjOJe.gif?fit=540%2C540&ssl=1","thumbnail_width":540,"thumbnail_height":540,"description":"Mathematicians including Newton,&nbsp;Leibniz, the&nbsp;Bernoulli family,&nbsp;Riccati,&nbsp;Clairaut,&nbsp;d&#8217;Alembert, and&nbsp;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&nbsp;Newton&#8217;s second law&nbsp;of motion \u2014 the relationship between the displacement&nbsp;x&nbsp;and the time&nbsp;t&nbsp;of an object under the force&nbsp;F, is given by the differential equation, which constrains the motion of a particle of constant mass&nbsp;m. This equation is probably the most popular Ordinary Differential Equation. Definition 1.1&#8211; A differential equation involving derivatives with respect to a single independent variable is called an ordinary differential equation. Linear and non-linear differential equations&#8211; Definition 1.2&#8211; 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&#8211; Definition 1.3&#8211; 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&#8211; Definition 1.4&#8211; 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. TYPES OF SOLUTIONS Let F (x, y, y1, y2, \u2026, yn) = 0 \u2026&#8230;&#8230;.. (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&#8211; 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. \u2026&#8230; (1)Differentiating (1) w.r.t. \u2018x\u2019, we get dy\/dx = me^mx. \u2026&#8230; (2)Now, (1) and (2) , dy\/dx = my , m = (1\/y) \u00d7 (dy\/dx). \u2026&#8230; (3)Again, from (1), mx = ln y so that m = (ln y)\/x. \u2026&#8230; (4)Eliminating m from (3) and (4), we get (1\/y) \u00d7 (dy\/dx) = (1\/x) \u00d7 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 isy = mx, m being arbitrary constant. \u2026&#8230; (1)Differentiating (1) w.r.t. \u2018x\u2019, dy\/dx = m. \u2026&#8230; (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 byy = mx + c, m and c being arbitrary constants. \u2026&#8230; (1)Differentiating (1) w.r.t. \u2018x\u2019, we get dy\/dx = m. \u2026&#8230; (2)Differentiating (2) w.r.t. \u2018x\u2019, we get d2y\/dx2 = 0, \u2026&#8230; (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 \u2018tspan\u2019 zv=0.1:0.01:0.5; % Vector Of \u2018z\u2019 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 \u2018gs\u2019 In \u2018gs2\u2019 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), \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. If, however, the identity implies that c1 = c2 = \u2026 = cn = 0, then y1, y2, \u2026, yn are said to be linearly independent. Existence and uniqueness theorem Consider a second order linear differential equation of the form a0 (x) y&#8221; + a1 (x) y&#8217; + 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&#8217; (x0) = c2. Moreover, this solution y (x) is defined over the interval (a, b)."}