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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 15EQ

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Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear
equation. The most important such equation has the form
y' + p(t)y = q(t)y"
and is called Bernoulli's equation after Jakob Bernoulli.
If n
0, 1, then the substitution v = yl-n reduces Bernoulli's equation to a linear equation.
Solve the given Bernoulli equation by using this substitution.
ty' + 4ty - y³ = 0, t > 0
y = ±
+ ct8
y = ±
O
O
O
O
O
+1
y = ±
y = ±
2
+ Ct8
5t
+ ct4
1
+ ct4
+ ct
5t
9t

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Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equation has the form y + p(t)y = q(1)y" and is called Bernoulli's equation after Jakob Bernoulli. If n # 0, 1, then the substitution v = y¹-" reduces Bernoulli's equation to a linear equation. Solve the given Bernoulli equation by using this substitution. t²y + 8ty-y² = 0,t> 0 y = = + +4₂¹6 y = ± +C1² y = ± +ql6 y = ± y = ± O O O O 17t 1 √9 2 + c18 1 +c¹6 17t
Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equation has the form y + p(t)y = q(t)y" and is called Bernoulli's equation after Jakob Bernoulli. If n + 0, 1, then the substitution v = y'-* reduces Bernoulli's equation to a linear equation. Solve the given Bernoulli equation by using this substitution. Py + 2ty – y = 0, t >0 1 y = + + ct V 51 5t +cf 3t y3+ 1 y =+ + cr² V
2. (LI-2) Consider the equation r²y" - xy + y = 0. It is given that y₁ = z is one solution to this equation. Use reduction of order to find a second solution 92 which is linearly independent from y₁.
In the xy-plane, the graph of a linear equation of the form y = mx + b and the graph of an exponential equation of the form y = ab* both contain points (1, 3) and (2, 4). If the point (r, s) is on the graph of the linear equation and the point (r, t) is on the graph of the exponential equation, where 0 t, which of the following must be true? (A) 0
In what way is the specific solution for the equation sought?
Find the general solution of the given equation y" + y = 0
is it possible to get this answer in that equation?
The presence of nonlinear terms prevents us from usingthe technique of this section. In special cases, a changeof variable will transform the nonlinear equation into onethat is linear. The equation known as Bernoulli’s equation,x' = a(t)x + f (t)x^n, n /= 0, 1,was proposed for solution by James Bernoulli in December 1695. In 1696, Leibniz pointed out that the equationcan be reduced to a linear equation by taking x 1−n as thedependent variable. Show that the change of variable,z = x^1−n, will transform the nonlinear Bernoulli equationinto the linear equationz' = (1 − n)a(t)z+(1−n)f(t)
I already posted this problem but would like a more thorough explanation I am completely lost still.
Find a particular solution to the given higher-order equation. y''' - y'"' - 22y' + 40y = - 40t² +244t-268 A particular solution is yp (t) = .

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