In 124 find the general solution of the given

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. dy dx 5y

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. dy dx 2y 0

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. dy dx  y e3x

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. 3 dy dx 12y 4

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. y 3x2y x2

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. y 2xy x3

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. x2y xy 1

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. y 2y x2 5

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. x  2y 3 dy dx  y x2 sin x

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. x dy dx x  2y 3

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. x dy dx  4y x3 x

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. (1  x) dy dx xy x x2

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. x2 y x(x 2)y ex

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. xy (1 x)y ex sin 2x

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. y dx 4(x y6 ) dy 0

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. y dx (yey 2x) dy

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. cos x dy dx  (sin x)y 1

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. cos2 x sin x dy dx  (cos3 x)y 1

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. (x 1) dy dx (x 2)y 2xex

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. (x 2)2 dy dx 5 8y 4xy

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. dr d r sec cos 

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. dP dt 2tP P 4t 2

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. x dy dx (3x 1)y e3x

In Problems 124 find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the general solution. (x2 1) dy dx 2y (x 1)2

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. dy dx x 5y, y(0) 3 (

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. dy dx 2x 3y, y(0) 1 3

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined . xy y ex , y(1) 2

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. y dx dy x 2y2 , y(1) 5 Di

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. L di dt Ri E, i(0) i0, L, R, E, i0 constants

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. dT dt k(T Tm), T(0) T0, k, Tm, T0 constants

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. x dy dx y 4x 1, y(1) 8

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. y 4xy x3 ex2 , y(0) 1

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. (x 1) dy dx y ln x, y(1) 10

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. x(x 1)dy dx xy 1, y(e) 1

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. y (sin x)y 2 sin x, y( >2) 1

In Problems 2536 solve the given initial-value problem. Give the largest interval I over which the solution is defined. y (tan x)y cos2 x, y(0) 1

In Problems 3740 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). 3 dy dx 2y f(x), y(0) 0,

In Problems 3740 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). dy dx y f(x), y(0) 1,

In Problems 3740 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). dy dx 2xy f(x), y(0) 2, f

In Problems 3740 proceed as in Example 6 to solve the given initial-value problem. Use a graphing utility to graph the continuous function y(x). (1 x2 ) dy dx 2xy f(x), y(0) 0,

Proceed in a manner analogous to Example 6 to solve the initial-value problem y P(x)y 4x, y(0) 3, where Use a graphing utility to graph the continuous function y(x).

Consider the initial-value problem y ex y f(x), y(0) 1. Express the solution of the IVP for x 0 as a nonelementary integral when f(x) 1. What is the solution when f(x) 0? When f(x) ex ? 43.

Express the solution of the initial-value problem y 2xy 1, y(1) 1, in terms of erf(x).

Reread the discussion following Example 2. Construct a linear first-order differential equation for which all nonconstant solutions approach the horizontal asymptote y 4 as x : .

Reread Example 3 and then discuss, with reference to Theorem 1.2.1, the existence and uniqueness of a solution of the initial-value problem consisting of xy 4y x6 ex and the given initial condition. (a) y(0) 0 (b) y(0) y0, y0 0 (c) y(x0) y0, x0 0, y0 0 46

Reread Example 4 and then find the general solution of the differential equation on the interval (3, 3).

Reread the discussion following Example 5. Construct a linear first-order differential equation for which all solutions are asymptotic to the line y 3x 5 as x : .

Reread Example 6 and then discuss why it is technically incorrect to say that the function in (9) is a solution of the IVP on the interval [0, ).

(a) Construct a linear first-order differential equation of the form xy a0(x)y g(x) for which yc cx3 and yp x3 . Give an interval on which y x3 cx3 is the general solution of the DE. (b) Give an initial condition y(x0) y0 for the DE found in part (a) so that the solution of the IVP is y x3 1x3 . Repeat if the solution is y x3 2x3 . Give an interval I of definition of each of these solutions. Graph the solution curves. Is there an initial-value problem whose solution is defined on , )? (c) Is each IVP found in part (b) unique? That is, can there be more than one IVP for which, say, y x3 1x3 , x in some interval I, is the solution? 50. In dete

In determining the integrating factor (3), we did not use a constant of integration in the evaluation of P(x) dx. Explain why using P(x) dx c1 has no effect on the solution of (2)

Suppose P(x) is continuous on some interval I and a is a number in I. What can be said about the solution of the initial-value problem y P(x)y 0, y(a) 0?

Radioactive Decay Series The following system of differential equations is encountered in the study of the decay of a special type of radioactive series of elements: where 1 and 2 are constants. Discuss how to solve this system subject to x(0) x0, y(0) y0. Carry out your ideas.

Heart Pacemaker A heart pacemaker consists of a switch, a battery of constant voltage E0, a capacitor with constant capacitance C, and the heart as a resistor with constant resistance R. When the switch is closed, the capacitor charges; when the switch is open, the capacitor discharges, sending an electrical stimulus to the heart. During the time the heart is being stimulated, the voltage E across the heart satisfies the linear differential equation Solve the DE, subject to E(4) E0.

(a) Express the solution of the initial-value problem y 2xy 1, , in terms of erfc(x). (b) Use tables or a CAS to find the value of y(2). Use a CAS to graph the solution curve for the IVP on (, ).

(a) The sine integral function is defined by Si(x) , where the integrand is x 0 (sin t>t) dt y(0) 1 2 dE dt 1 RC E. defined to be 1 at t 0. Express the solution y(x) of the initial-value problem x3y 2x2y 10 sin x, y(1) 0 in terms of Si(x). (b) Use a CAS to graph the solution curve for the IVP for x 0. (c) Use a CAS to find the value of the absolute maximum of the solution y(x) for x 0. 56. (a

(a) The Fresnel sine integral is defined by . Express the solution y(x) of the initial-value problem y (sin x2)y 0, y(0) 5, in terms of S(x). (b) Use a CAS to graph the solution curve for the IVP on (, ). (c) It is known that S(x) : as x : and S(x) :  as x : . What does the solution y(x) approach as x : ? As x : ? (d) Use a CAS to find the values of the absolute maximum and the absolute minimum of the solution y(x).