Question

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 ) = sqrt(2)*y ( x ) at x = π/4.

(The square root in the right hand side

of the equation takes positive values and − 1 ≤ y ≤ 1)

c)

Let y be the solution of the equation

y ′ + (3x^2y)/(1+x^3)=(e^x)/(1+x^3) satisfying the condition y ( 0 ) = 1.

Find ln ( 2 y ( 1 ) ).

Answer #1

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...

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...

Let f(x, y) = x tan(xy^2) + ln(2y). Find the equation of the
tangent plane at (π, 1⁄2).

Verify that the function
y=x^2+c/x^2
is a solution of the differential equation
xy′+2y=4x^2, (x>0).
b) Find the value of c for which the solution satisfies the initial
condition y(4)=3.
c=

Find the solution to the separable differential equation dy =
x cos2 y + sin x cos2 y satisfying π dx
the initial condition y = 4 when x = π.

Let f(x, y) = sqrt( x^2 − y − 4) ln(xy).
• Plot the domain of f(x, y) on the xy-plane.
• Find the equation for the tangent plane to the surface at the
point (4, 1/4 , 0).
Give full explanation of your work

1. Find the point on the curve y = x2 that is closest to (0,
5).
2. Find the function f(x),iff′′(x)=sinx+x and f(0)=f(π)=0.
3. Find derivatives of the following functions. a) arcsin(
square root 3x)

Let X and Y have a joint density function given by f(x; y) = 3x;
0 <= y <= x <= 1
(a) Find P(X<2Y).
(b) Find cov(X,Y).
(c) Find P(X < 1/2 |Y = 1/3).
(d) Find P(X = 1/2|Y = 1/3).
(e) Find P(X > 1/2|Y > 1/3).
(f) Find the conditional expectation E(X|Y = y).

Let y=2−3x+∑n=2∞an x power n be the power series solution of the differential equation:
y″+6xy′+6y=0 about x=0. Find a4.

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

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