Taylor Expansion and Derivative Formulas for Matrix Logarithms
I give the derivation of formulas for the Taylor expansion and derivative of a matrix logarithm. log(x + y) - log(x) - log(x + y + U) + log(x + U) =.
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“Covariates impacts in compositional models and simplicial
2 dic 2019 logarithm is on the right hand side of the regression equation) or symmetrically the partial derivative of E(log(Y )) with respect to X in ...
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“Covariates impacts in compositional models and simplicial
2 dic 2019 logarithm is on the right hand side of the regression equation) or symmetrically the partial derivative of E(log(Y )) with respect to X in ...
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DIFFERENTIAL EQUATIONS
Now substituting x = 1 in the above
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Impact of covariates in compositional models and simplicial derivatives
25 mar 2021 with respect to log(X) (if the logarithm is on the right hand side of the regression equation) or symmetrically the partial derivative of ...
Impact of Covariates in Compositional Models and S
On a Linear Differential Equation of the Second Order
zero and consequently the fundamental integrals of the equation in the domain of x = O are Yi = ii (x)
CONTINUITY AND DIFFERENTIABILITY
Then we say logarithm of a to base b is x if bx=a log. = x b. 6. logb b = 1 and logb 1 = 0. (iv) The derivative of ex w.r.t.
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Differential Equations Assignment #1: answers.
One quickly checks that x(t) = log(t) · et2 is a solution of the equation and satisfies the initial condition; it is even unique as such by the Cauchy-
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Dimensions of Logarithmic Quantities
note that eq 1 may be written in a differential form: (x + d log (x) = log (x + dje) - tion multiplying each equation by the stoichiometric num-.
A REFINEMENT OF SELBERG'S ASYMPTOTIC EQUATION
ψ(x) - Σ Λ{n) - Σ log p R{x)= ψ(x) - x we have the two equivalent forms of Selberg's asymptotic equation. (1). R(x)log x + J*Λ(-
) df(t) = O(x)
9.1 Overview
(i) An equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation. (ii) A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation and a differential equation involving derivatives with respect to more than one independent variables is called a partial differential equation. (iii) Order of a differential equation is the order of the highest order derivative occurring in the differential equation. (iv) Degree of a differential equation is defined if it is a polynomial equation in its derivatives. (v) Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it. (vi) A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution. (vii) To form a differential equation from a given function, we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants. (viii) The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves. (ix) 'Variable separable method" is used to solve such an equation in which variables can be separated completely, i.e., terms containingxshould remain withdxand terms containingy should remain withdy.Chapter 9DIFFERENTIAL EQUATIONS
180 MATHEMATICS(x) A function F (x,y) is said to be a homogeneous function of degreen if
F (lx, ly)= ln F (x,y) for some non-zero constantl. (xi) A differential equation which can be expressed in the formdy dx= F (x,y) ordx dy = G (x,y), where F (x,y) and G (x,y) are homogeneous functions of degree zero, is called a homogeneous differential equation. (xii) To solve a homogeneous differential equation of the typedy dx = F (x,y), we make substitutiony =vx and to solve a homogeneous differential equation of the typedx dy = G (x,y), we make substitutionx =vy. (xiii) A differential equation of the formdy dx + Py = Q, where P and Q are constants or functions ofx only is known as a first order linear differential equation. Solution of such a differential equation is given byy (I.F.) =( )Q I.F.dx´ò+ C, whereI.F. (Integrating Factor) =Pdxeò.
(xiv) Another form of first order linear differential equation isdx dy + P1x = Q1, where P1 and Q1 are constants or functions ofy only. Solution of such a differential
equation is given byx (I.F.) =( )1Q × I.F.dyò+ C, where I.F. =1Pdyeò.9.2 Solved Examples
Short Answer (S.A.)
Example 1Find the differential equation of the family of curvesy = Ae2x + B.e-2x.Solutiony = Ae2x + B.e-2x
DIFFERENTIAL EQUATIONS 181dy
dx = 2Ae2x - 2 B.e-2x and2 2d y dx = 4Ae2x + 4Be-2x Thus2 2d y dx = 4yi.e.,2 2d y dx- 4y =0. Example 2Find the general solution of the differential equationdy dx=y x.Solutiondy
dx=y xÞdy y =dx xÞdy yò=dx xòÞ logy = logx + logcÞ y =cxExample 3Given thatdy
dx=yex andx = 0,y =e.Find the value ofy whenx = 1.Solutiondy
dx=yex Þ dy yò =xe dxò Þlogy =ex +c Substitutingx = 0 andy =e,we get loge= e0+ c, i.e., c = 0 (loge = 1)Therefore, logy= ex.
Now, substitutingx = 1 in the above, we get logy = eÞ y=ee.Example 4Solve the differential equationdy
dx +y x=x2. SolutionThe equation is of the type+P = Qdyydx, which is a linear differential equation.Now I.F. =1dxxò=elogx =x.
Therefore, solution of the given differential equation is182 MATHEMATICSy.x =2xx dxò, i.e. yx =4
4xc+Hencey =3
4x c x+. Example 5Find the differential equation of the family of lines through the origin. SolutionLety =mx be the family of lines through origin. Therefore,dy dx=mEliminatingm, we gety =dy
dx.x or xdy dx -y = 0. Example 6Find the differential equation of all non-horizontal lines in a plane. SolutionThe general equation of all non-horizontal lines in a plane is ax +by =c, wherea¹0.Therefore,dxa bdy+= 0.
Again, differentiating both sides w.r.t.y, we get22d xady = 0Þ2
9.1 Overview
(i) An equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation. (ii) A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation and a differential equation involving derivatives with respect to more than one independent variables is called a partial differential equation. (iii) Order of a differential equation is the order of the highest order derivative occurring in the differential equation. (iv) Degree of a differential equation is defined if it is a polynomial equation in its derivatives. (v) Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it. (vi) A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution. (vii) To form a differential equation from a given function, we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants. (viii) The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves. (ix) 'Variable separable method" is used to solve such an equation in which variables can be separated completely, i.e., terms containingxshould remain withdxand terms containingy should remain withdy.Chapter 9DIFFERENTIAL EQUATIONS
180 MATHEMATICS(x) A function F (x,y) is said to be a homogeneous function of degreen if
F (lx, ly)= ln F (x,y) for some non-zero constantl. (xi) A differential equation which can be expressed in the formdy dx= F (x,y) ordx dy = G (x,y), where F (x,y) and G (x,y) are homogeneous functions of degree zero, is called a homogeneous differential equation. (xii) To solve a homogeneous differential equation of the typedy dx = F (x,y), we make substitutiony =vx and to solve a homogeneous differential equation of the typedx dy = G (x,y), we make substitutionx =vy. (xiii) A differential equation of the formdy dx + Py = Q, where P and Q are constants or functions ofx only is known as a first order linear differential equation. Solution of such a differential equation is given byy (I.F.) =( )Q I.F.dx´ò+ C, whereI.F. (Integrating Factor) =Pdxeò.
(xiv) Another form of first order linear differential equation isdx dy + P1x = Q1, where P1 and Q1 are constants or functions ofy only. Solution of such a differential
equation is given byx (I.F.) =( )1Q × I.F.dyò+ C, where I.F. =1Pdyeò.9.2 Solved Examples
Short Answer (S.A.)
Example 1Find the differential equation of the family of curvesy = Ae2x + B.e-2x.Solutiony = Ae2x + B.e-2x
DIFFERENTIAL EQUATIONS 181dy
dx = 2Ae2x - 2 B.e-2x and2 2d y dx = 4Ae2x + 4Be-2x Thus2 2d y dx = 4yi.e.,2 2d y dx- 4y =0. Example 2Find the general solution of the differential equationdy dx=y x.Solutiondy
dx=y xÞdy y =dx xÞdy yò=dx xòÞ logy = logx + logcÞ y =cxExample 3Given thatdy
dx=yex andx = 0,y =e.Find the value ofy whenx = 1.Solutiondy
dx=yex Þ dy yò =xe dxò Þlogy =ex +c Substitutingx = 0 andy =e,we get loge= e0+ c, i.e., c = 0 (loge = 1)Therefore, logy= ex.
Now, substitutingx = 1 in the above, we get logy = eÞ y=ee.Example 4Solve the differential equationdy
dx +y x=x2. SolutionThe equation is of the type+P = Qdyydx, which is a linear differential equation.Now I.F. =1dxxò=elogx =x.
Therefore, solution of the given differential equation is182 MATHEMATICSy.x =2xx dxò, i.e. yx =4
4xc+Hencey =3
4x c x+. Example 5Find the differential equation of the family of lines through the origin. SolutionLety =mx be the family of lines through origin. Therefore,dy dx=mEliminatingm, we gety =dy
dx.x or xdy dx -y = 0. Example 6Find the differential equation of all non-horizontal lines in a plane. SolutionThe general equation of all non-horizontal lines in a plane is ax +by =c, wherea¹0.Therefore,dxa bdy+= 0.
Again, differentiating both sides w.r.t.y, we get22d xady = 0Þ2
- log x differentiation formula