A First Course in Differential Equations with Modeling Applications (11th Edition)

Chapter 6

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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The interval of convergence is and the radius of ... more

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; The Maclaurin Series of the function is and ... more

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; The Maclaurin Series of the function is and ... more

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; The Maclaurin Series of the function is and ... more

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; The Maclaurin Series of the function is and ... more

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; The Maclaurin Series of the function is and ... more

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; The Maclaurin Series of the function is and ... more

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; The Taylor Series of the function is and its ... more

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; The Taylor Series of the function is and its ... more

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; The Maclaurin series of is and the Maclaurin ... more

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; The Maclaurin Series of the function is .... more

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  ; The Maclaurin series of is . And also ... more

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; The Maclaurin series of is and also .... more

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; Let hence . Determine the upper and lower ... more

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; Let hence . Determine the upper and lower ... more

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; Let .Substitute for in the expression to ... more

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; Let .Substitute for in the expression to ... more

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; Let .Substitute for in the expression to ... more

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; Substitute for in the expression to ... more

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; Substitute for in the expression to ... more

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; Let in the first and third summation and in ... more

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Differentiate the equation with respect to to ... more

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Differentiate the equation with respect to to ... more

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Differentiate the equation with respect to to ... more

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Differentiate the equation with respect to to ... more

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; Differentiate the equation with respect to to... more

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; Differentiate the equation with respect to to... more

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; Differentiate the equation with respect to to... more

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; Differentiate the equation with respect to to... more

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; Multiply and divide the function by 2 to ... more

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The interval in which the Maclaurin series for ... more

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For , For , ; For the differential equation , and... more

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For , For , ; For the differential equation , and... more

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; Differentiate the equation with respect to to... more

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The solutions and are similar to each other and ... more

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The solutions and are similar to each other and ... more

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The solutions and are similar to each other and ... more

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; Let the solution be . Determine the first and ... more

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  ; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Consider .Substitute for , for and for in... more

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; Consider .Substitute for , for and for in... more

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; Consider .The series expansion for is given as... more

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; Consider .The expansion series for is given by... more

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About ; About ; ; The equation can be written ... more

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Yes ; Comparing the differential equation to the ... more

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We say that a point x=a is an ordinary point for ... more

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The differential equation can be written as .... more

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We rewrite the differential equation in the ... more

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Singular points : 0, and Regular singular points... more

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Singular points : 0, 5Regular singular points: 0, ... more

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Singular points : -3, 2Regular singular points: -3... more

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Singular points : 0, , Regular singular points: 0... more

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Irregular singular points : 0Regular singular ... more

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Irregular singular points : and Regular singular ... more

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For , and For , and ; The differential equation... more

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For , and ; The differential equation can be ... more

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and -1Two series solution ; The differential ... more

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1 and 0One series solution ; The differential ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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; Let the solution be . Determine the first and ... more

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  ; Let the solution be . Determine the first and ... more

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  ; Let the solution be . Determine the first and ... more

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The differential equation is analytic at regular ... more

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Yes, the equation corresponds to the Cauchy-Euler... more

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; Compare the differential equation with ... more

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; Compare the differential equation with ... more

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  ; Divide the given equation by 4 on both sides ... more

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  ; Simplify the differential equation to obtain ... more

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; Multiply both sides of the equation with... more

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; The given equation is,. Simplify the equation... more

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; The given equation is , which has the singular ... more

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; The given equation is,. Which has the singular ... more

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; The given the following differential equation ... more

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; The given equation as, has the form . Compare ... more

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; Determine the first and second derivative of ... more

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; The given equation is and . Differentiating ... more

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; The given equation is . Put it in the form ... more

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; The given equation is . Put it in form  Divide ... more

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; The given equation is . Put it in form Divide ... more

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; The given equation is . Put it in the ... more

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  ; The given equation is Put it in the ... more

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; The given equation is Put it in the form ... more

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; The given equation is Put it in the form ... more

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The given equation is Also, two equations to be ... more

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; Simplify the differential equation to obtain ... more

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; Simplify the differential equation to obtain ... more

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; Simplify the differential equation to obtain ... more

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; Substitute for in the equation and simplify ... more

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Substitute 1 for in the equation to obtain the ... more

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Substitute 0 for in the equation to obtain the ... more

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; Substitute for in the equation to obtain the... more

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; Substitute for in the equation to obtain the... more

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Differentiate the equation with respect to to ... more

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; The root of the differential equation is and... more

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For eigenvalue , the eigenfunction is .Foe ... more

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; The equation of recurrence relation is given by... more

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Differentiate the equation two times with respect... more

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, , and ; Compare the differential equation with ... more

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Let , then and .Substitute for , for , and for... more

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; Let , then and .Substitute for , for , and ... more

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Differentiate the equation with respect to to ... more

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; The Rodrigues’ formula for Legendre polynomials... more

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For ; For ; For ; For ; For ; For ; For ; Insert ... more

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; ; ; ; ; ; ;The zeros of higher order can be ... more

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The differential equation is the  Bessel equation... more

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It is given that is an irregular singular point ... more

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; Simplify the equation to find in terms of ... more

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; Substitute for and for in the equation to ... more

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; Consider a differential equation .Rewrite the ... more

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; Let .As is a regular singular point so, and .... more

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; Let .Differentiate the equation  with respect ... more

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; Let .Differentiate the equation with respect ... more

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; Let .Differentiate the equation with respect ... more

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; Let .Differentiate the equation with respect ... more

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; Let .Differentiate the equation with respect ... more

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; Let .Differentiate the equation with respect ... more

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; Let .Differentiate the equation with respect ... more

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; The lower bound of the radius of convergence of... more

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; Let .As the initial conditions are given at the... more

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0 is an ordinary point. ; The given differential ... more

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0 is a regular singular point. ; The given ... more

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; Let .Differentiate the equation with respect ... more

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