Questions and Answers of A First Course in Differential Equations

In problem first use (18) to express the general solution of the given differential equation in terms of Bessel functions. Then use (23) and (24) to express the general solution in terms of
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
In problem use the procedure in Example 8 to find two power series solutions of the given differential equation about the ordinary point x = 0.y'' + (sin x)y = 0
The first-order differential equation dy/dx = x2 + y2 cannot be solved in terms of elementary functions. However, a solution can be expressed in terms of Bessel functions.(a) Show that the
Assume that b in equation (18) can be pure imaginary, that is, b = βi, > 0, i2 = -1. Use this assumption to express the general solution of the given differential equation in terms the modified
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
In problem use the power series method to solve the given initial-value problem.(x2 + 1)y'' + 2xy' = 0, y(0) = 0, y'(0) = 1
Use a substitution to shift the summation index so that the general term of given power series involves xk. nc, n+2 n=1
In problem the given function is analytic at a = 0. Use appropriate series in (2) and long division to find the first four nonzero terms of the Maclaurin series of the given function.
Note that x = 0 is an ordinary point of the differential equation y'' + x2y' + 2xy = 5 2x + 10x3. Use the assumption
Use the series in (7) to verify that Iυ (x) = i-υ Jυ (ix) is a real function.
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
In problem use the power series method to solve the given initial-value problem.y'' – 2xy' + 8y = 0, y(0) = 3, y'(0) = 0
In problem the given function is analytic at a = 0. Use appropriate series in (2) and long division to find the first four nonzero terms of the Maclaurin series of the given function.sec x
In problem investigate whether x = 0 is an ordinary point, singular point, or irregular singular point of the given differential equation. Recall the Maclaurin series for cos x and ex.(ex - 1 - x)y''
In problem use (18) to find the general solution of the given differential equation on (0, ∞).9x2y'' + 9xy' + (x6 - 36)y = 0
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
In problem use the power series method to solve the given initial-value problem.(x + 1)y'' – (2 – x)y' + y = 0, y(0) = 2, y'(0) = -1
In problem the given function is analytic at a = 0. Use appropriate series in (2) and multiplication to find the first four nonzero terms of the Maclaurin series of the given function.e-xcos x
In problem investigate whether x = 0 is an ordinary point, singular point, or irregular singular point of the given differential equation. Recall the Maclaurin series for cos x and ex.xy'' + (1 - cos
In problem use (18) to find the general solution of the given differential equation on (0, ∞).xy'' + 3y' + x3y = 0
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
In problem use the power series method to solve the given initial-value problem.(x – 1)y'' – xy' + y = 0, y(0) = -2, y'(0) = 6
In problem the given function is analytic at a = 0. Use appropriate series in (2) and multiplication to find the first four nonzero terms of the Maclaurin series of the given function.sin x cos x
Even though x = 0 is an ordinary point of the differential equation, explain why it is not a good idea to try to find a solution of the IVPy'' + xy' + y = 0, y(1) = -6, y'(1) = 3of the form
In problem use (18) to find the general solution of the given differential equation on (0, ∞).4x2y" + (16x2 + 1)y = 0
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
Find two power series solutions of the given differential equation about the ordinary point x = 0.(x2 - 1)y" + xy' - y = 0
Use an appropriate series in (2) to find the Taylor series of the given function centered at the indicated value of a. Write your answer in summation notation.ln x; a = 2; (x = 2[1 + (x - 2)/2])
Use an appropriate series in (2) to find the Taylor series of the given function centered at the indicated value of a. Write your answer in summation notation.sin x, a = 2π Use periodicity.
Without actually solving the differential equation (1 - 2 sin x)y'' + x y = 0, find a lower bound for the radius of convergence of power series solutions about the ordinary point x = 0.
In problem use (18) to find the general solution of the given differential equation on (0, ∞).x2y" + (x2 - 2)y = 0
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
Find two power series solutions of the given differential equation about the ordinary point x = 0.(x2 + 2)y" + 3xy' - y = 0
Use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation.sin x2
Solve the given initial-value problem.(x + 2)y'' + 3y = 0, y(0) = 0, y'(0) = 1
In problem use (18) to find the general solution of the given differential equation on (0, ∞).xy" - 5y' + xy = 0
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
Find two power series solutions of the given differential equation about the ordinary point x = 0.(x2 + 1)y' - 6y = 0
Use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation.ln(1 x)
Solve the given initial-value problem.y'' + xy' + 2y = 0, y(0) = 3, y'(0) = -2
In problem use (18) to find the general solution of the given differential equation on (0, ∞).xy" - y' + xy = 0
In problem, x = 0 is a regular singular point of the given differential equation. Show that the indicial roots of the singularity do not differ by an integer. Use the method of Frobenius to obtain
Find two power series solutions of the given differential equation about the ordinary point x = 0.y"- (x + 1)y' - y = 0
Use an appropriate series in (2) to find the Maclaurin series of the given function. Write your answer in summation notation.x/1 + x2
In problem use an appropriate infinite series method about x 0 to find two solutions of the given differential equation.(cos x)y" + y = 0
In problem use (18) to find the general solution of the given differential equation on (0, ∞).xy" + 3y' + xy = 0
In problem, x = 0 is a regular singular point of the given differential equation. Use the general form of the indicial equation in (14) to find the indicial roots of the singularity. Without solving,
Find two power series solutions of the given differential equation about the ordinary point x = 0.(x + 2)y" + xy' - y = 0
In problem use an appropriate infinite series method about x 0 to find two solutions of the given differential equation.xy"- (x + 2)y' + 2y = 0
In problem use (18) to find the general solution of the given differential equation on (0, ∞).xy" + 2y' + 4y = 0
In problem, x = 0 is a regular singular point of the given differential equation. Use the general form of the indicial equation in (14) to find the indicial roots of the singularity. Without solving,
Find two power series solutions of the given differential equation about the ordinary point x = 0.(x - 1)y'' + y' =0
In problem use an appropriate infinite series method about x 0 to find two solutions of the given differential equation.y" - x2y' + xy = 0
In problem use the indicated change of variable to find the general solution of the given differential equation on (0, ∞).x2y'' + (α2x2 – υ2 + 1/4)y = 0; y = √x v(x)
In problem put the given differential equation into form (3) for each regular singular point of the equation. Identify the functions p(x) and q(x).xy'' + (x + 3)y' + 7x2y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0.y'' + 2xy' + 2y = 0
In problem use an appropriate infinite series method about x 0 to find two solutions of the given differential equation.(x - 1)y" + 3y = 0
In problem use the indicated change of variable to find the general solution of the given differential equation on (0, ∞).x2y'' + 2xy' + α2x2y = 0; y = x-1/2 v(x)
In problem put the given differential equation into form (3) for each regular singular point of the equation. Identify the functions p(x) and q(x).x2 1)y'' + 5(x + 1)y' + (x2 x)y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0.y'' + x2y' + xy = 0
In problem use an appropriate infinite series method about x 0 to find two solutions of the given differential equation.y" - xy' - y = 0
In problem use (12) to find the general solution of the given differential equation on (0, ∞).x2y'' + xy' + (2x2 64)y = 0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.(x3 - 2x2 + 3x)2y" + x(x - 3)2y' - (x + 1)y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0.y'' - xy' + 2y = 0
In problem use (12) to find the general solution of the given differential equation on (0, ∞).x2y' + xy' + (25x2 4/9)y = 0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.x3(x2 - 25)(x - 2)2y" + 3x(x - 2)y' + 7(x + 5)y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0.y" - 2xy' + y = 0
Regular singular points at x = 1 and at x = -3In problem construct a linear second-order differential equation that has the given properties.
In problem use (12) to find the general solution of the given differential equation on (0, ∞).x2y'' + xy' + + (36x2 ¼)y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0.y'' + x2y = 0
A regular singular point at x = 1 and an irregular singular point at x = 0In problem construct a linear second-order differential equation that has the given properties.
In problem use (12) to find the general solution of the given differential equation on (0, ∞).x2y'' + xy' + (9x2 4)y = 0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.(x2 + x - 6)y" + (x + 3)y' + (x - 2)y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0.y" - xy = 0
Use the Maclaurin series for sin x and cos x along with long division to find the first three nonzero terms of a power series in x for the function f(x) = sinx/cosx.
In problem use (1) to find the general solution of the given differential equation on (0, ∞). d [xy'] + (x dx 4 =D0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.x2(x - 5)2y" + 4xy' + (x2 - 25)y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equations obtained using the
Suppose the power series Σ∞k=0ck(x – 4)k is known to converge at -2 and diverge at 13. Discuss whether the series converges at 7, 0, 7, 10, and 11. Possible answers are does, does not, might.
In problem use (1) to find the general solution of the given differential equation on (0, ∞).xy'' + y' + xy = 0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.(x3 + 4x)y" - 2xy' + 6y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equations obtained using the
x = 0 is an ordinary point of a certain linear differential equation. After the assumed solution y = Σ∞n=0 cnxn is substituted into the DE, the following algebraic system is obtained by equating
In problem use (1) to find the general solution of the given differential equation on (0, ∞).16x2y'' + 16xy' + (16x2 1)y = 0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.y" – 1/x y' + 1/(x - 1)3y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equations obtained using the
Both power series solutions of y'' + ln(x + 1)y' + y = 0 centered at the ordinary point x = 0 are guaranteed to converge for all x in which one of the following intervals?(a) (∞,∞)(b) (1,
In problem use (1) to find the general solution of the given differential equation on (0, ∞).4x2y'' + 4xy' + (4x2 25)y = 0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.(x2 - 9)2y" + (x + 3)y' + 2y = 0
Find two power series solutions of the given differential equation about the ordinary point x = 0. Compare the series solutions with the solutions of the differential equations obtained using the
Because x = 0 is an irregular singular point of x3y'' – xy' + y = 0, the DE possesses no solution that is analytic at x + 0. _________In problem answer true or false without referring back to the
In problem use (1) to find the general solution of the given differential equation on (0, ∞).x2y'' + xy' + (x2 1)y = 0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.x(x + 3)2y" - y = 0
In problem without actually solving the given differential equation, find the minimum radius of convergence of power series solutions about the ordinary point About the ordinary point x = 1.(x2 2x
The general solution of x2y'' + x y' + (x2 - 1)y = 0 is y = c1J1(x) + c2 J-1(x). _____In problem answer true or false without referring back to the text.
In problem use (1) to find the general solution of the given differential equation on (0, ∞).x2y'' + xy' + + (x2 1/9y) = 0
In problem determine the singular points of the given differential equation. Classify each singular point as regular or irregular.x3y" + 4x2y' + 3y = 0
In problem without actually solving the given differential equation, find the minimum radius of convergence of power series solutions about the ordinary point About the ordinary point x = 1.(x2
Show that the amplitude of the steady-state current in the LRC-series circuit in Example 10 is given by E0/Z, where Z is the impedance of the circuit.

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