Question

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 be the solution of the equation y ′ = 4 x + y^2

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.

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 ″ − 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...

Consider the differential equation y'=-x-y Use Euler's method
deltax=.1 to estimate y when x=0.2 for the solution curve
satisfying y(-1)=0: Euler's approximation give y(0.2)=?

Consider the following initial value problem:
dy/dt = -3 - 2 *
t2, y(0) = 2
With the use of Euler's method, we would like to find an
approximate solution with the step size h = 0.05 .
What is the approximation of y
(0.2)?

Use Euler's method with step size 0.1 to estimate y(0.5), where
y(x) is the solution of the initial-value problem
y'=3x+y^2, y(0)=−1
y(0.5)=

dy/dx = x^4/y^2
initial condition y(1)= 1
a) use eulers method to approximate the solution at x=1.6 and
a step size od delta x = 0.2
b) solve the differential equation exactly using seperation
variabled and the intial condtion y(1)=1.
c) what is the exact value of y(1.6) for your solution from
part b.

(3)If H(x, y) = x^2 y^4 + x^4 y^2 + 3x^2 y^2 + 1, show that H(x,
y) ≥ 0 for all (x, y). Hint: find the minimum value of H.
(4) Let f(x, y) = (y − x^2 ) (y − 2x^2 ). Show that the origin
is a critical point for f which is a saddle point, even though on
any line through the origin, f has a local minimum at (0, 0)

a)Program a calculator or computer to use Euler's method to
compute y(1), where y(x) is the solution
of the given initial-value problem. (Give all answers to four
decimal places.)
dy
dx
+ 3x2y =
9x2,
y(0) = 4
h =
1
y(1) =
h =
0.1
y(1) =
h =
0.01
y(1) =
h =
0.001
y(1) =
(b) Verify that
y = 3 +
e−x3
is the exact solution of the differential equation.
y = 3 +
e−x3
⇒ y'...

Apply Euler's method twice to approximate the solution of the
equation y'=y-x-1, y(0)=1 at x=0.5. Use h=0.1.
a.
y(0.5)=1.089
b.
y(0.5)=0.579
c.
y(0.5)=1.534
d.
y(0.5)=0.889

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.

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