Question

a)

Let y be the solution of the equation y ″ − y = 3 e^(2x)

satisfying the conditions y ( 0 ) = 2 and y ′ ( 0 ) = 3.

Find the value of the function f ( x ) = ln ( y ( x ) − e^x ) at x = 3.

b)

Let y be the solution of the equation y ″ − 2 y ′ + y = x − 2

satisfying the conditions y ( 0 ) = 0 and y ′ ( 0 ) = 2.

Find the value of the function f ( x ) = y ( x ) − x e^x at x = 3.

Answer #1

Let y be the solution of the equation
a)
y ′ = 2 x y, satisfying the condition y ( 0 ) = 1.
Find the value of the function f ( x ) = ln ( y ( x ) )
at the point x = 2.
b)
Let y be the solution of the equation
y ′ = sqrt(1 − y^2) satisfying the condition y ( 0 )
= 0.
Find the value of the function f ( x...

a)
Let y be the solution of the equation
y ′ − [(3x^2*y)/(1+x^3)]=1+x^3 satisfying the
condition y ( 0 ) = 1.
Find y ( 1 ).
b)
Let y be the solution of the equation y ′ = 4 − 2 x y
satisfying the condition y ( 0 ) = 0.
Use Euler's method with the horizontal step size h =
1/2
to find an approximation to the value of the function
y at x = 1.
c)
Let y...

B. a non-homogeneous differential equation, a complementary
solution, and a particular solution are given. Find a solution
satisfying the given initial conditions.
y''-2y'-3y=6 y(0)=3 y'(0) = 11 yc=
C1e-x+C2e3x
yp = -2
C. a third-order homogeneous linear equation and three linearly
independent solutions are given. Find a particular solution
satisfying the given initial conditions
y'''+2y''-y'-2y=0, y(0) =1, y'(0) = 2, y''(0) = 0
y1=ex, y2=e-x,,
y3= e-2x

Let f(x, y) = c/x, 0 < y < x < 1 be the joint density
function of X and Y .
a) What is the value of c?
a) 1 b) 2 c) 1/2 d) 2/3 e) 3/2
b)what is the marginal probability density function of X?
a) x/2 b)1 c)1/x d)x e)2x
c)what is the marginal probability density function of Y ?
a) ln y b)−ln y c)1 d)y e)y2
d)what is E[X]?
a)1 b)2 c)4 d)1/2 e)1/4

Find the solution to the linear system of differential
equations
{x′ = 6x + 4y
{y′=−2x
satisfying the initial conditions x(0)=−5 and
y(0)=−4.
x(t) = _____
y(t) = _____

Consider the first order separable equation
y′=12x^3y(1+2x^4)^1/2. An implicit general solution can be written
in the form y=Cf(x) for some function f(x) with C an arbitrary
constant. Here f(x)= Next find the explicit solution of the initial
value problem y(0)=1
y=

12. Let f(x, y) = c/x, 0 < y < x < 1 be the joint
density function of X and Y . What is
the value of c?
a) 1; b) 2; c) 1/2; d) 2/3; e) 3/2.
13. In Problem 12, what is the marginal probability density
function of X?
a) x/2; b)1; c)1/x; d)x; e)2x.
14. In Problem 12, what is the marginal probability density
function of Y ?
a) ln y; b)−ln y; c)1; d)y; e)y2....

Use Newton-Raphson to find a solution to the polynomial equation
f(x) = y where y = 0 and
f(x) = x^3 + 8x^2 + 2x
- 40. Start with x(0) = 1 and continue until (6.2.2) is
satisfied with e= 0.0000005.

Let F ( x , y ) = 〈 e^x + y^2 − 3 , − e ^(− y) + 2 x y + 4 y 〉.
a) Determine if F ( x , y ) is a conservative vector field and, if
so, find a potential function for it. b) Calculate ∫ C F ⋅ d r
where C is the curve parameterized by r ( t ) = 〈 2 t , 4 t + sin
π...

Consider the function f(x,y) = (e^{2x})lny whose domain is
{(x,y): y>0}. What is the equation of the plane
tangent to the surface z=f(x,y) at (x,y) = (3,5)?

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