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Q5E

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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 450

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

In Problems 1-10, use a substitution y=x^r to find the general solution to the given equation for x>0.
d²y/dx²=5/x dy/dx-13/x² y

The general solution for the given equation is y=c₁x³ cos (2 lnx)+c₂x³ sin(2 lnx).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Define Cauchy-Euler equations:

In mathematics, a Cauchy problem is one in which the solution to a partial differential equation must satisfy specific constraints specified on a hypersurface in the domain.

An initial value problem or a boundary value problem is both examples of Cauchy problems.

The equation will be in the form of ax²y"+bxy'+cy=0.

Find the general solution:

The given equation is

d²y/dx²=5/x dy/dx-13/x² y

This can be re-written in the required form as

x²y"-5xy'+13y=0

Let L be the differential operator defined by the left-hand side of equation, that is

L[y](x) = x²y"-5xy'+13y

And let's set,

w(r,x)=x^r

Substituting the w(r,x) in place of y(x), you get

L[y](x) = x²(x^r )"-5x(x^r)'+13(x^r)

=x² (r(r-1)) x^r-2 -5x(r) x^r-1 +13x^r

=(r²-r) x^r -5rx^r+13x^r

=(r²-6r+13) x^r

Solving the indicial equation,

r²-6r+13=0

r=[2± √(36-4×13)]/2

=(6± 4i)/2

=3±2i

There are two complex conjugates,

r₁=3+2i and r₂=3-2i

Thus there are two linearly independent solutions given by,

x³⁺²ⁱ=x³ cos(2 lnx)+ix³ sin (2 lnx)

You can write two linearly independent real-valued solutions as,

y₁=x³ cos(2 lnx) and y₂=x³ sin(2 lnx)

Therefore, the general solution for the equation will be,

y=c₁ x³ cos(2 lnx)+c₂ x³ sin(2 lnx)

Most popular questions for Math Textbooks

In Problems 15-17,solve the given initial value problem x³y"'+6x²y"+29xy'-29y=0 y(1)=1 and y'(1)= -3 and y"(1)=19.

In Problems 21-28, use the procedure illustrated in Problem 20 to find at least the first four nonzero terms in a power series expansion about’s x=0 of a general solution to the given differential equation.

(1+x²)y"-xy'+y=e^-x

The solution to the initial value problem

$\begin{array}{l} x y^{''} (x) + 2 y^{'} (x) + x y (x) = 0 \\ y (0) = 1, y^{'} (0) = 0 \end{array}$

has derivatives of all orders at $x = 0$ (although this is far from obvious). Use L'Hôpital's rule to compute the Taylor polynomial of degree 2 approximating this solution.

Find a minimum value for the radius of convergence of a power series solution about x₀.

(x²-5x+6) y"-3xy'-y=0; x₀=0

Question: In Problems 1–10, determine all the singular points of the given differential equation.

7. (sinx)y"+(cosx)y =0

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