1.Use Euler's Method to approximate the solution to at t=1 using eight steps.

b) Do the same with the improved Euler method and compare results.

2.Solve the initial value problem

3.Find the general solution for the Bernoulli differential equation:

4.Solve the initial value problem

5.Use a power series expansion about x=0 to find a general solution for: (Determine at least the first four nonzero terms.)

6.A damped vibrating spring under an external driving force can be modeled by the equation:

where m>0 is the mass of the spring, b is the damping constant, k>0 the spring constant and g(t) the driving force. If y(t) is the displacement from equilibrium at time t, determine the form(general solution) of the equation of motion if (assume ). What is the behavior of the solution as ?

7.Solve the initial value problem .

8.Solve the problem using matrix methods, that is:

Convert the equation to a system of first order linear equations---that is, a matrix equation of the form: where

9.A 12 hour water clock (clepsydra) is designed to the dimesions shown in the figure, where the shape of the surface is obtained by revolving the curve around the y-axis. What should this curve be and what should the radius of the (circular) bottom hole be so that the water rate falls at a constant rate of 4 inches per hour? [Hint: The rate of change of the vertical axis y is governed by Torricelli's Law, which leads to the equation: where A(y) is the cross-sectional area at height y, and a is the (constant) area of the drain hole at the bottom, and g is the gravatational acceleration (constant). Note that these latter two constants can be combined into the constant k.]

10.Solve the initial value problem:given that is one particualr solution of the equation.

11.Find the general solution in powers of x of the differential equation given below. State the resursion relation and radius of convergence.

12.Plot a phase plane diagram for the system:

Final Review Set (Summer 2017)- Math 246 Show all your work for each problem x ' ( t )=1+t sin ( tx ) , x ( 0 ) =0 at t=1 using eight steps. 1 1 1 1 8 1 b) Do the same with the improved x 2= + 1 + sin 8 8 8 8 1. A) Use Euler's Method to approximate the solution to ( ( )) 1 1 x 1=x 0+ hf (t 0 , x 0) 1 8 x 1=0+ 1+ sin 8 8 Euler method and compare results. x j+1 =x j + ( ( ))( ) h 1+ x j sin ( x j y j ) +1+ x j +1 sin x j+1 ( y j+h ( 1+ x j sin ( x j y j ) ) ) 2 ( ( )) ( y j+h )+ x j sin ( x j y j ) ( x j+ h)() ( ) 2+ x j sin ( x j y j ) +( x j+b )sin h x j +1=x j + 2 1 +0 8 2 () 8 ( ) 1 1 2 2+ sin 0 + sin 8 8 8 1 x1=0+ 16 () ( )( ) 0.1250 () 3 1 2 ( +0.1250+ sin ) 8 8 8 () 1 0.12502 3 2+ sin + sin 8 8 8 1 x 2=0.1250+ 16 ( )( ) 3 0.25004 8 ( ) 4 1 3 ( +0.25004+ sin ) 8 8 8 ( ) 1 0.12503 4 2+ sin + sin 8 8 8 1 x3 =0.25004+ 16 ( )() ( 2 2. Solve the initial value problem tan ( y)2+ x sec ( y )+ ( )( ) 1 ' y ( x )=0; y ( 0 ) =1 y ) 3. Find the general solution for the Bernoulli differential equation: 4. Solve the initial value problem 4 0.375180 8 () 5 1 4 ( + 0.375180+ sin) 8 8 8 ( ) 1 0.3751804 5 2+ sin + sin 8 8 8 1 x 4=0.375180+ 16 ' y ( x ) +2 y= y ' ' ( x )+ 6 y' ( x )+5 y ( x )=0, y ( 0 )=1, y ' ( 0 ) = x y2 1 3 5. Use a power series expansion about x=0 to find a general solution for: ( 2 x3 ) y ' ' x y' + y =0 (Determine at least the first four nonzero terms.) 6. A damped vibrating spring under an external driving force can be modeled by the equation: m y ' ' + b y ' + ky=g(t ) where m>0 is the mass of the spring, b is the damping constant, k>0 the spring constant and g(t) the driving force. If y(t) is the displacement from equilibrium at time t, determine the form (general solution) of the equation of motion if g ( t )=sin ( t) (assume b2