Consider the initial value problemry +y = -2ry², y(1) = -7.(a) This differential equation can be written in the form (*) withP(z) =Q(z) =and%3Dn =(b) The substitution u =will transform it into the linear equationduU =dr(C) Using the substitution in part (b), we rewrite the initial condition in terms of r and uu(1)(d) Now solve the linear equation in part (b), and kad the solution that satisfies the initial condition in part (c).u(x) =(e) Finally, solve for y.y(x) = YA Bernoulli differential equation is one of the formdy+ P(x)y= Q(r)y"(*)deObserve that, if n = 0 or 1, the Bermoulli equation is linear. For other values of n, the substitution u = y transforms the Bernoulli equationinto the linear equationdu+ (1 – n)P(x)u = (1= n)Q(x).daConsider the initial value problem.xy + y = -2ay², y(1) = -7.(a) This differential equation can be written in the form (*) withP(x)Q(z) =andn =(b) The substitution uwill transform it into the linear equationduda(C) Using the substitution in part (b), we rewrite the initial condition in terms of x and uu(1)

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A Bernoulli differential equation is one of the form dy + P(x)y = Q(x)y" (*) dz %3D Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y-n transforms the Bernoulli equat into the linear equation du + (1 – n)P(x)u = (1 – n)Q(r). da Consider the initial value problem. xy +y=-2æy, y(1) = –7. (a) This differential equation can be written in the form (*) with P(x) Q(z) and n = (b) The substitution u = will transform it into the linear equation du da (c) Using the substitution in part (b), we rewrite the initial condition in terms of r and u u(1)

A Bernoulli differential equation is one of the form dy + P(x)y = Q(x)y" () dz %3D Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u = y-n transforms the Bernoulli equat into the linear equation du + (1 – n)P(x)u = (1 – n)Q(r). da Consider the initial value problem. xy +y=-2æy, y(1) = –7. (a) This differential equation can be written in the form () with P(x) Q(z) and n = (b) The substitution u = will transform it into the linear equation du da (c) Using the substitution in part (b), we rewrite the initial condition in terms of r and u u(1)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Consider the initial value problem
ry +y = -2ry², y(1) = -7.
(a) This differential equation can be written in the form (*) with
P(z) =
Q(z) =
and
%3D
n =
(b) The substitution u =
will transform it into the linear equation
du
U =
dr
(C) Using the substitution in part (b), we rewrite the initial condition in terms of r and u
u(1)
(d) Now solve the linear equation in part (b), and kad the solution that satisfies the initial condition in part (c).
u(x) =
(e) Finally, solve for y.
y(x) =

YA Bernoulli differential equation is one of the form
dy
+ P(x)y= Q(r)y"
(*)
de
Observe that, if n = 0 or 1, the Bermoulli equation is linear. For other values of n, the substitution u = y transforms the Bernoulli equation
into the linear equation
du
+ (1 – n)P(x)u = (1= n)Q(x).
da
Consider the initial value problem.
xy + y = -2ay², y(1) = -7.
(a) This differential equation can be written in the form (*) with
P(x)
Q(z) =
and
n =
(b) The substitution u
will transform it into the linear equation
du
da
(C) Using the substitution in part (b), we rewrite the initial condition in terms of x and u
u(1)

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A Bernoulli differential equation is one of the form dydx+P(x)y=Q(x)yn.dydx+P(x)y=Q(x)yn. Observe that, if n=0n=0 or 11, the Bernoulli equation is linear. For other values of nn, the A Bernoulli differential equation is one of the form dydx+P(x)y=Q(x)yn.dydx+P(x)y=Q(x)yn. Observe that, if n=0n=0 or 11, the Bernoulli equation is linear. For other values of nn, the substitution u=y1−nu=y1−n transforms the Bernoulli equation into the linear equation dudx+(1−n)P(x)u=(1−n)Q(x).dudx+(1−n)P(x)u=(1−n)Q(x). Use an appropriate substitution to solve the equation y′−(5/x)y=y^4/x^10,y′−((5/x)y)=y^4/x^13, and find the solution that satisfies y(1)=1.y(1)=1.
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