Find the general solution. Indicate which method in this chapter you are using. Show the details of your work. y'+2.5y=1.6x

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y'=6(y-2.5)tanh 1.5x

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.)

$$\text{xy'+4y=8}x^4\text{, y(1)=2}$$

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.)

$$\text{y'+y sin x=}e^{cosx},\text{ y(0)=-2.5}$$

Solve the IVP. Indicate the method used. Show the details of your work. 3 sec y dx+1/3 sec x dy=0, y(0)=0

Solve the differential equations

$$1 - \quad x y ^ { \prime } - 2 y = x ^ { 3 } e ^ { x }$$

$$2 - \quad ( 2 y d x + d y ) e ^ { 2 x } = 0$$

Find the motion of the mass–spring system modeled by the ODE and the initial conditions. Sketch or graph the solution curve. In addition, sketch or graph the curve of y-yp to see when the system practically reaches the steady state. (D²+8D+17l)y=474.5 sin 0.5t, y(0)=-5,4, y'(0)=9,4

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y'-y=5.2

Using a method of this section or separating variables, find the general solution. If an initial condition is given, find also the particular solution and sketch or graph it. y'=(tan y)/(x-1), y(0)=½π

Solve the ODE by integration or by remembering a differentiation formula

$$y'+2 \sin(2πx)=0$$

Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an initial condition is given, find the corresponding particular solution. 2xtanydx+sec²ydy=0

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.) y'+2y=4cos2x, y(1/4π)=3

Find the general solution. If an initial condition is given, find also the corresponding particular solution and graph or sketch it. (Show the details of your work.)

$$y'+ky=e^{-kx}$$

Find a general solution. Show the steps of derivation. Check your answer by substitution. xy'=y²+y (Set y/x=u)