Question

Transform the differential equation x^{2}d^{2}y/
dx^{2} − 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.

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...

Find the
i)particular integral of the following differential equation
d2y/dx2+y=(x+1)sinx
ii)the complete solution of d3y /dx3-
6d2y/dx2 +12 dy/dx-8 y=e2x
(x+1)

Use the method for solving homogeneous equations to solve the
following differential equation.
(9x^2-y^2)dx+(xy-x^3y^-1)dy=0
solution is F(x,y)=C, Where C= ?

Find the particular integral of the differential equation
d2y/dx2 + 3dy/dx + 2y = e −2x
(x + 1). show that the answer is yp(x) = −e −2x ( 1/2
x2 + 2x + 2)

(61). (Bernoulli’s Equation): Find the general solution of the
following first-order differential equations:(a) x(dy/dx)+y=
y^2+ln(x) (b) (1/y^2)(dy/dx)+(1/xy)=1

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

Use the z-transform method to solve the following difference
equation: y[n + 2) = 3y[n + 1] – 2y[n], y[0] = 5, y[1] = 0)

Use the substitution x = et to transform the given Cauchy-Euler
equation to a differential equation with constant coefficients.
(Use yp for dy /dt and ypp for d2y/dt2 .) x2y'' + 10xy' + 8y =
x2
Solve the original equation by solving the new equation using
the procedures in Sections 4.3-4.5. y(x) =

1) Solve the given differential equation by separation of
variables.
exy
dy/dx = e−y +
e−6x −
y
2) Solve the given differential
equation by separation of variables.
y ln(x) dx/dy = (y+1/x)^2
3) Find an explicit solution of the given initial-value
problem.
dx/dt = 7(x2 + 1), x( π/4)= 1

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 .

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