Question

1) Solve the given differential equation by using an appropriate substitution. The DE is a Bernoulli equation.

x dy/dx +y= 1/y^2

2)Consider the following differential equation.

(25 − y^{2})y' = x^{2}

Let f(x, y) = x^2/ 25-y^2. Find the derivative of *f*.
af//ay=

Determine a region of the *xy*-plane for which the given
differential equation would have a unique solution whose graph
passes through a point

(x_{0}, y_{0}) in the region.

a) A unique solution exists in the region consisting of all
points in the *xy*-plane except (0, 5) and (0, −5).

b) A unique solution exists in the regions y < −5, −5 < y < 5, and y > 5.

c) A unique solution exists in the region y > −5.

d) A unique solution exists in the entire *xy*-plane.

e)A unique solution exists in the region y < 5.

Answer #1

solve the given differential equation by using an
appropriate substitution. The DE is a Bernoulli equation. x * dy/dx
+ y = 1/y^2

Consider the differential equation
y' =
y2 − 9
.
Let
f(x, y) =
y2 − 9
.
Find the partial derivative of f.
df
dy
=
Determine a region of the xy-plane for which the given
differential equation would have a unique solution whose graph
passes through a point
(x0, y0)
in the region.
A unique solution exits in the entire x y-plane.
A unique solution exists in the region −3 < y < 3.
A unique solution exits...

Solve the given differential equation by using an appropriate
substitution. The DE is of the form dy dx = f(Ax + By + C), which
is given in (5) of Section 2.5. dy/dx = (x + y + 7)^2

determine if the xy-plane for which the given differential
equation would have a unique solution whose graph passes through
the point (x0,y0) in the region
dy/dx=y^(2/3)
x(dy/dx)=y

Solve the Bernoulli differential equation. The Bernoulli
equation is a well-known nonlinear equation of the form
y' + P(x)y = Q(x)yn
that can be reduced to a linear form by a substitution. The
general solution of a Bernoulli equation is
y1 − ne∫(1 − n)P(x) dx =
(1 − n)Q(x)e∫(1 − n)P(x) dx dx
+ C.
(Enter your solution in the form F(x, y) = C or y = F(x, C)
where C is a needed constant.)
y' − 10y...

Solve the Bernoulli differential equation. The Bernoulli
equation is a well-known nonlinear equation of the form
y' + P(x)y = Q(x)yn
that can be reduced to a linear form by a substitution. The
general solution of a Bernoulli equation is
y1 − ne∫(1 − n)P(x) dx =
(1 − n)Q(x)e∫(1 − n)P(x) dx dx
+ C.
(Enter your solution in the form F(x, y) = C or y = F(x, C)
where C is a needed constant.)
y8y' − 5y9...

Determine a region of the xy-plane for which the given
differential equation would have a unique solution whose graph
passes through a point (x_0, y_0) in the region. (1+y^3)y' =
x^2

A Bernoulli differential equation is one of the form
dy/dx+P(x)y=Q(x)y^n (∗)
Observe that, if n=0 or 1, the Bernoulli equation is linear. For
other values of n, the substitution u=y^(1−n) transforms the
Bernoulli equation into the linear equation
du/dx+(1−n)P(x)u=(1−n)Q(x).
Consider the initial value problem xy′+y=−8xy^2, y(1)=−1.
(a) This differential equation can be written in the form (∗)
with P(x)=_____, Q(x)=_____, and n=_____.
(b) The substitution u=_____ will transform it into the linear
equation du/dx+______u=_____.
(c) Using the substitution in part...

Consider the Bernoulli equation dy/dx + y = y^2, y(0) = −1
Perform the substitution that turns this equation into a linear
equation in the unknown u(x).
Solve the equation for u(x) using the Laplace transform.
Obtain the original solution y(x). Does it sound familiar?

(a) verfiy that y=tan(x+c) ia a one parameter family of
solutions of the differential equation y'= 1+x^2
(b)since f(x,y)= 1+y^2 and df/dy= 2y are continuous everywhere,
the region R can be taken to be the entire xy-plane. Use the family
of solutions in part A to find an explicit solution of the first
order initial value problem y'= 1+y^2, y(0)=0. Even though x0=0 is
in the interval (-2,2) explain why the solution is not defined on
its interval
(c)determine the...

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