An equation of the form y' + a(x) y = b(x)y is called a Bernoulli equation,...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
An equation of the form y' + a(x) y = b(x)y" is called a Bernoulli equation, named for Jakob Bernoulli (1654-1705), one of a family of noted Swiss scientists and mathematicians. Note that if n = 0 or n = 1, we have a linear equation. Now if n is not equal to 0 or 1 and we divide both sides of the equation by y", we can let ,1-n and get a linear equation in the variable z. We solve the linear equation for z in terms of x and then return to the original variables x and y. This substitution method was found by Leibniz in 1696. For example, y'- y = xy² is a Bernoulli equation with a(x) = −1, b(x) = x, and n = 2. Divide by y2 and the equation be- comes y-² y'- y¹ = x. Letting z = y¹, we get the linear equation -z' -z = x, or z' + z = -x. Solving for z, we find that z = 1 - x + ce*. Since z = =y-¹, we conclude that y = (1-x+ce¯)-¹. Note that we divided by y² and that y = 0 is a singular solution. Find solutions for each Bernoulli equation in Problem x = = x + √√x An equation of the form y' + a(x) y = b(x)y" is called a Bernoulli equation, named for Jakob Bernoulli (1654-1705), one of a family of noted Swiss scientists and mathematicians. Note that if n = 0 or n = 1, we have a linear equation. Now if n is not equal to 0 or 1 and we divide both sides of the equation by y", we can let ,1-n and get a linear equation in the variable z. We solve the linear equation for z in terms of x and then return to the original variables x and y. This substitution method was found by Leibniz in 1696. For example, y'- y = xy² is a Bernoulli equation with a(x) = −1, b(x) = x, and n = 2. Divide by y2 and the equation be- comes y-² y'- y¹ = x. Letting z = y¹, we get the linear equation -z' -z = x, or z' + z = -x. Solving for z, we find that z = 1 - x + ce*. Since z = =y-¹, we conclude that y = (1-x+ce¯)-¹. Note that we divided by y² and that y = 0 is a singular solution. Find solutions for each Bernoulli equation in Problem x = = x + √√x
Expert Answer:
Related Book For
Differential Equations and Linear Algebra
ISBN: 978-0131860612
2nd edition
Authors: Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West
Posted Date:
Students also viewed these accounting questions
-
An AC waveform has an equation of the form v = 2002 sin (314t + /3) volts. Calculate the circuit current in rectangular form if it flows through an impedance of 20 k.
-
In Exercises 1-4, find an equation of the line passing through the pair of points. Sketch the line. 1. (5, 1), (5, 5) 2. (4, 3), (4, 4) 3. (7, 2), (7, 5) 4. (6, 3), (2, 3)
-
In problem, find an equation of the line L. y. (2, 1) (0, 0) -2 -1 2 X
-
Since perpetuity payments continue forever, how can a present value be computed? Why isnt the present value infinite?
-
Why must a company estimate the amount of factory overhead assigned to individual jobs or job lots?
-
Let be a random process, where An and Bn are random variables such that , E[An]= E[Bn]= 0 E[AnBm] = 0 E[AnAm] dn, mE An = [ 2] E[BnBm] δn, mE [ Bn 2], , and for all and , where is the...
-
9. This case concerns unequal taxes on property. In Pennsylvania, a county tax commissioner appraises land, meaning that he sets a value for the land, and the owner then pays real estate taxes based...
-
Quaint Construction, Inc., is a home builder in Arizona. Quaint uses a job order costing system in which each house is a job. Because it constructs houses, the company uses an account titled...
-
Please use excel and show all formulas and do awritten explanationQuestion 1 & 2Need all excel and written explanation for#2 Stock Valuation at Ragan, Inc. Ragan, Inc., was founded nine years ago by...
-
Thaarugo,inc, produces a GPS device that is becoming popular in parts of Scandinavia. When Thaarugo produces one of these, a printed circuit board (PCB) is used and it is populated with several...
-
Tyler has heard a great deal about the differences between Windows and Mac-based software for graphics and artistic work. Now that you have two new Windows-based personal computers for the...
-
You measure the velocity of falling object at several times given in the table below. Time (s) Velocity (m/s) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 The acceleration is the slope of...
-
41. 42. The following table lists data collected during a recent experiment. In this case, x represents age (in years) and y represents the given diameter (in inches). 3 5 7 11 13 y 5.73 6.44 7.01...
-
Using the slope formula, compute the slope of the line that goes through (17, 5.5) and (16,-5). Slope 22.5/21 Not equivalent. Now find the equation of the line in the form = mx + b. Equation Nope....
-
A neutral lithium atom is ionized, losing an electron to form Li* and resulting in a decrease in its atomic size. The ionization energy required to do this is 520kJ/mol. If we were to remove a second...
-
A physical examination was used to scan for breast cancer in 2500 women with biopsy proven adenocarcinoma of the breast and 5000 age and race matched control women. The result of the physical...
-
A particle is moving along a straight line with an initial velocity of 6 m / s when it is subjected to a deceleration of a = ( 1 . 5 v ) m / s 2 , where v is in m / s . Part A How much time does this...
-
Maria Castigliani is head of the purchasing department of Ambrosiana Merceti, a medium-sized construction company. One morning she walked into the office and said, The main problem in this office is...
-
Some first-order DEs have periodic solutions or solutions that tend toward an oscillation. For the DEs in Problems, use isoclines to sketch a direction field and describe whatever sort of periodicity...
-
Lengths Show that the distance between the heads of any two vectors uÌ and vÌ, as shown in Fig. 3 . 1 .5, has length ||uÌ - vÌ ||. Ce FIGURE 3.1.5 Diagram for Problem 82
-
When a lint-order DE has the form y' = f(y), so the right-band side doesn't depend oo t, the equation is called autonomous (which means independent of time). The logistic equation y' = ky(1 - y) in...
-
Petronet sells merchandise to its 80-percnet subsidiary Sonata at a markup of 20 percent on cost. During 2017, Petronet charges Sonata \($4,000,000\) for merchandise sales. Sonatas 2017 beginning...
-
Pentamedia owns 90 percent of Sesa. At the start of 2016, Sesa sold buildings carried at \($4,000,000\), net, to Pentamedia for \($6,000,000\). The buildings had a remaining life of 10 years and...
-
A parent sells land costing \($35,000\) to a subsidiary in 2016 for \($55,000\). The subsidiary sells the land in 2018 to a third party for \($85,000\). On the consolidated income statement for 2018,...
Study smarter with the SolutionInn App