Question

A Bernoulli differential equation is one of the form dxdy+P(x)y=Q(x)yn

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)

Use an appropriate substitution to solve the equation y'−(3/x)y=y^4/x^2 and find the solution that satisfies y(1)=1

Answer #1

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?

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 following statements.
(i) The differential equation y′ + P(x) y = Q(x) has the form
of a linear differential equation.
(ii) All solutions to y′ = e^(sin(x^2 + y)) are increasing
functions throughout their domain.
(iii) Solutions to the differential equation y′ = f (y) may
have different tangent slope for points on the curve where y = 3,
depending on the value of x

Solve the differential equation:
x*(du(x)/dx)=[x-u(x)]^2+u(x)
Hint: Let y=x-u(x)
Answer: u(x)=x-x/(x-K) with K=constant

Use the method for solving Bernoulli equations to solve the
following differential equation
dx/dy+5t^7x^9+x/t=0
in the form F(t,x)=c

Solve the Homogeneous differential equation
(7 y^2 + 1 xy)dx - 1 x^2 dy = 0
(a) A one-parameter family of solution of the equation is y(x)
=
(b) The particular solution of the equation subject to the
initial condition y(1) =1/7.

Transform the differential equation x2d2y/
dx2 − xdy/dx − 3y = x 1−n ln(x), x > 0 to
a linear differential equation with constant coefficients. Hence,
find its complete solution using the D-operator method.

If y=∑n=0∞cnx^n is a solution of the differential
equation y′′+(x+1)y′−1y=0, then its coefficients cn are
related by the equation
cn+2= _______cn+1 +______ cn .

(1 point)
In this problem we consider an equation in differential form
Mdx+Ndy=0Mdx+Ndy=0.The equation
(4e−2y−(20x4y5e−x+2e−xsin(x)))dx+(−(20x5y4e−x+8e−2y))dy=0(4e−2y−(20x4y5e−x+2e−xsin(x)))dx+(−(20x5y4e−x+8e−2y))dy=0
in differential form M˜dx+N˜dy=0M~dx+N~dy=0 is not exact.
Indeed, we have
M˜y−N˜x=

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