Differential equations Questions and Answers

Consider this ordinary differential equation ODE dy dx At the point 1 5 0 5 the direction field for this ODE has a slope of At the point 1 5 0 5 the direction field for this ODE has a slope of Which of the following is a direction field for this ODE OL 2x 2y 11 X 1 11 1 1 1 2 2 1121 1X I 11 X HT 7 1 1 1 1 1

y 0 03y 9 y 0 140 s solution alt

The IVP 7yy x y 7 6 has solution y x

e separable IVP dy dx y x 0 9y with initial condition y 0 4 has solution

dy 73 dx x has solution aux y 4 MATRA THANKELES PARATER 1 32

The IVP y 14xy 2x y 0 19 has solution y x

Use the Product and Quotient Rules as necessary to find the derivative of f x 3x cos x Use symbolic notation and fractions where needed f x

h t 0 5t 4 a Sketch the graph v No Solution Help 10 9 8 7 6 5 4 3 2 1 b Find the slope of the graph 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 10 Fill WebAssign Graphing Tool Graph La After yo can use propertie

Problem 7 5 points Find the area of the region above in the z direction the curve t t tlnt t 0 1 t e and below the surface z 1 Fore a 3 such that fo

Draw six isoclines gradient curves for the differential equation x y and then make dy dx partial slope field by considering points on the isoclines

3 The plane passes through D 3 5 4 with normal 2 1 3 Determine the Scalar Form of the plane and the exact intercepts 5 marks

x t sint cos 2t y t cost sin 2t Find x axis rotate

3 Consider the following models of harmonic motion Y 5 sin 2t 1 y2 5 sin 2t 3 Do both motions have the same frequency What is the phase for each equation What is the phase difference Are the objects moving in phase or out of phase

Consider the initial value problem u t u t u t 24 0 5 Although the differential equation associated with this problem is non linear the differential equation for y t u t P where p linear Using this observation we can see that the solution to the given initial value problem is u t Number

One solution to an ordindary differential equation ODE is y 1 Which of the following direction fields could correspond to this ODE 2 12 X TI I 2VI Q Q 7 1 71 1 1 7 1 1 1 1 1 1121 II 1

Verify that for any value of c the function y ce 2 e is a solution to this ODE y 2y e then find the value of C for which the solution satisfies the initial condition y 3 8 C

Consider this ordinary differential equation ODE dy dz At the point 1 0 5 the direction field for this ODE has a slope of YTA At the point 1 5 0 5 the direction field for this ODE has a slope of Which of the following is a direction field for this ODE O 111 1 1 1 21 11 VA 7 1 1 17 1 1 7 12V 1 Q

11 2 1 dy dx dy dx 1 dy dx Which of the following equations would produce the direction field above dy dx x x y 1 x y x y 1 x y 1

If y x ek is a solution of this differential equation y 10y 24y 0 then the possible values of k are

The solution to this initial value problem dy dx is u r 6x 5 y 2 5

If r t k is a contant solution to this ordinary differential equation 8t dx dt then k 4x 4 0

oof Use an algebraic degree measure The measure

Problem 4 Find the radius of convergence of the following power series a b n 0 where d p n zn n 0 n in z n n p x a x n 0 is a polynomial c 8 m i 0 z n n n 2

Use Laplace transform to solve the ODE f t f t u t 1 with initial condition f 0 1 where u t is the Heaviside ste function

By directly applying the Cauchy Riemann equations show the following functions are analytic a f z z C 0 b f z ze c f z sin z

2 Let f x 3cosx 4sinx 5 TT a Determine the exact slope of f x at a 4 b Determine the tangent line approximation to y f x at the point where a c At the point where a is f x increasing decreasing or neither Why

Find 1 10 if f x x x 8 10 Simplify your answer

Find the power series solution for the differential equation y 3xy y 0 with the initial conditions y 0 2 a y 0 0 5 A y x 1 2 B x 11 72 6 x 5 11 4 6 y x x x 12 72 5 4 y 2 x x 12 11 72 11 4 6 y 2 x x x

Set up but do not evaluate an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line begin array c y 6 x x 2 y x operatorname about x 8 int 0 1 x end array

Consider the following 1 y a Find y by implicit differentiation y 6 X 7 b Solve the equation explicitly for y and differentiate to get y in terms of x y c Check that your solutions to parts a and b are consistent by substituting the expression for y into your solutio part a

Find the function y y x for x 0 which satisfies the separable differential equation dy 4 16x dx xy with the initial condition y 1 4 y x 0

dy Solve the separable differential equation 0 2y dx and find the particular solution satisfying the initial condition y 0 6 y x

Find a function y x such that 3yy x and y 3 11 y HR

29 Suppose that y is a solution to the initial value problem y y2 1 e y and y 1 0 Show that 1 y t 1 for all t for which y is defined

Obtain the differential equations with the following General Solution Co2x

1 Classify each of the following equations as to order and degree a 3dy 4xy secx dx 0 b L d q dt R dq dt q C E t c y xy 5y y x y 0 d d r dt 2 d r dt rt 6t e 2y 3y y 3cosx 8 Order Degree 11

If s1 2 and sn 1 p2 sn for n 1 2 3 prove that sn converges and that sn 2 for n 1 2 3

1 point Sometimes a change of variable can be used to convert a differential equation y y into a separable quation Une con technique is as follows 1 Consider a differential equation of the form y f ot By y where a 3 andy are constants Use the change of variable at By 7 to rewrite the differential equation as a separable equation of the form g z Solve the initial value problem a a 22 b 1 help formulas 3 1 2 5 of vange help formulas

Compare the graph of the function with the graph of f x arcsin x g x arcsin x 2 L 3 1 2 3 A 4 3 y I 2 4 A 4 KIN 2 3A 4 f x f x 1 3 KIN RI 4 RI RIN 2 37 st 1 A g x 2 g x 1 3 2 4 F 3 X X 2 3 1 2 1 37 4 1 KIN KIT 4 KIKIN 4 3 2 4 f x g x 1 3A RIN KIT 4 KIT KIN I 2 3 4 g x 2 f x 1 3 2 4 3

Area of a quadrilateral Given a quadrilateral that has vertices x y x2 Y 2 X3 Y3 and x4 y4 given in a counter clockwise order the area enclosed by the quadrilateral is given by X1 Y2 X3 1X4 A x 2 2 1 3 3 1 34 Y4 Y3 X4 V4 x3 Y3 3 1 Determine the area of the kite whose vertices are given by 2 0 4 3 2 4 and 0 3 where units of length are measured in inches 8 pts

ar Inequalities K The following graph shows the total wid in millions in the United States for the y idgets millions 10 8 6 United States Widget Production 11 4 7 7 Parti CIT 9 9 8 5 5 4 8 5 10 6

Consider the boundary value problem defined by the partial differentia equation in x t 8 6 2 6 x x y and the conditions Let o x y 8 6 dy 4 24y for 0 x 1 and 0 y 1 o 0 y 2y 3 1 y y 1 x 0 3 o x 1 2 for 0 y 1 for 0 x 1 V v y y x y where v y satisfies the partial differentia 3 and v 1 2 Find v y and re express the above boundary value problem in terms of x y equation above and the conditions v 0 Noto Do NOT alire the boundom roluo problem 1

Save ularity The quartic function f x 0 002x 4 0 025x 3 0 364x 2 7 243x 86 993 d to estimate the number of dolls of this type that were sold in thousands during a

Which of the following functions there may be more than one are solutions of the differential equation y 4y 4y et Oy e t Oy et y te t et Oy e t et y e t tet

question below Graph the circle with equation x 2 y 2 25 Plot the center and then a point on the circle 6 5 4 3 2 Clear AIL Draw 5 4 3 2 1 1 2 3 4 5 6 1 2 3 4 5 6

In the diagram below QN NP 2 3 and QP 10 Find the radius of 0 P 13 Q2 R 14 S Enter a number only

Find d y dx d y dx in terms of x and y x y 10x 2 3x 14 10xy 2 25 4 2 x y X

COOOOOOD similar polygons scale factors dilation enlargement reduction isometry right triangle legs of right triangle hypotenuse QPythagorean Theorem angle of elevation depression trigonometric ratio sine cosine tangent identity The polygons are are not similar because The relationship s between are Pythagorean Identity indirect measure Students may utilize Desmos or GraphNCalc 83 1 Are the two polygons similar Explain your reasoning C T13 Mod AS 5 2 5 cm 2 In AABC M and N are midpoints of sides What relationship s exist between segments MN and AC C T14 Mod AS 2 1 3 cm 140 130 2 1 cm 4 5 cm A 0 0 R 5 2 cm B 2 3 140 and 4 2 5 cm 8 7 cm 130 3 3 cm C 12 0

Determine a region of the xy plane for which the given differential equation would have a unique solution whose graph passes through a point xo yo in the region G dy d x x y

a Fill in the blanks with the coefficient f x x6x4 2 x 1 x 1 CO C1 C2 Hint Use Taylor s Theorem to find the Taylor series of the function f at x 1 You must show detailed work for the calculation of each coefficient b Use the binomial series to expand the function as a power series 1 x co c x c x 3x x x 1 x 1 4 C3 Cn Must show a compact nice form of Cn Strongly recommend that you do not combine calculate the exact value of C1 C2 C3 so that you can see the pattern for Cn