Given a second order linear homogeneous differential equation a2(x)y+a1(x)y'+a0(x)y=0 a 2 (x) y + a...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Given a second order linear homogeneous differential equation a2(x)y"+a1(x)y'+a0(x)y=0 a 2 (x) y " + a 1(x) y' + a0 (x) y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions y1,y2 y 1, y 2. But there are times when only one function, call it y1 y 1, is available and we would like to find a second linearly independent solution. We can find y2 y 2 using the method of reduction of order. First, under the necessary assumption the a2(x)#0 a 2 ( x ) #0 we rewrite the equation as y"+p(x)y'+q(x)y=0 p(x)=a1(x)a2(x), q(x)=a0(x)a2(x), y" +p(x) y'+q (x)y=0p( x) = a1 (x) a 2 (x), q (x) = a0 (x) a 2 (x), Then the method of reduction of order gives a second linearly independent solution as y2(x)=Cylu=Cy1(x) [e-Sp(x)dxy21(x)dx y 2 (x) = Cylu=Cy1(x) [e- Sp(x) dxy12(x) dx where C C is an arbitrary constant. We can choose the arbitrary constant to be anything we like. One useful choice is to choose C C so that all the constants in front reduce to 11. For example, if we obtain y2=C3e2x y 2 = C 3 e 2 x then we can choose C=1/3 C = 1/3 so that y2-e2x y 2 = e 2 x. Given the problem 25y"-20y'+4y=0 25 y" - 20 y' + 4y = 0 and a solution y1=e(2x/5) y 1 = e( 2x/5). Applying the reduction of order method to this problem we obtain the following y21(x)= y 12 (x) = p(x)= p (x) = and e-Sp(x)dx= e-S p(x) dx = So we have fe-Sp(x)dxy21(x)dx=[[e-Sp(x) dx y 12 (x) dx = [dx= d x = Finally, after making a selection of a value for C C as described above (you have to choose some nonzero numerical value) we arrive at y2(x)=Cylu= y 2 ( x ) = Cy1u = So the general solution to 25y"-20y'+4y=0 25 y" - 20 y' + 4y = 0 can be written as y=c1y1+c2y2=c1y = c1y 1+ c2y2=c1+c2 + c 2 Given a second order linear homogeneous differential equation a2(x)y"+a1(x)y'+a0(x)y=0 a 2 (x) y " + a 1(x) y' + a0 (x) y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions y1,y2 y 1, y 2. But there are times when only one function, call it y1 y 1, is available and we would like to find a second linearly independent solution. We can find y2 y 2 using the method of reduction of order. First, under the necessary assumption the a2(x)#0 a 2 ( x ) #0 we rewrite the equation as y"+p(x)y'+q(x)y=0 p(x)=a1(x)a2(x), q(x)=a0(x)a2(x), y" +p(x) y'+q (x)y=0p( x) = a1 (x) a 2 (x), q (x) = a0 (x) a 2 (x), Then the method of reduction of order gives a second linearly independent solution as y2(x)=Cylu=Cy1(x) [e-Sp(x)dxy21(x)dx y 2 (x) = Cylu=Cy1(x) [e- Sp(x) dxy12(x) dx where C C is an arbitrary constant. We can choose the arbitrary constant to be anything we like. One useful choice is to choose C C so that all the constants in front reduce to 11. For example, if we obtain y2=C3e2x y 2 = C 3 e 2 x then we can choose C=1/3 C = 1/3 so that y2-e2x y 2 = e 2 x. Given the problem 25y"-20y'+4y=0 25 y" - 20 y' + 4y = 0 and a solution y1=e(2x/5) y 1 = e( 2x/5). Applying the reduction of order method to this problem we obtain the following y21(x)= y 12 (x) = p(x)= p (x) = and e-Sp(x)dx= e-S p(x) dx = So we have fe-Sp(x)dxy21(x)dx=[[e-Sp(x) dx y 12 (x) dx = [dx= d x = Finally, after making a selection of a value for C C as described above (you have to choose some nonzero numerical value) we arrive at y2(x)=Cylu= y 2 ( x ) = Cy1u = So the general solution to 25y"-20y'+4y=0 25 y" - 20 y' + 4y = 0 can be written as y=c1y1+c2y2=c1y = c1y 1+ c2y2=c1+c2 + c 2
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
-
When a second-order linear homogeneous differential equation is written in the form (2.3), any point at which P(x) or Q(x) becomes infinite is called a singular point or singularity. In solving a...
-
2. (a) Consider the third order linear homogeneous differential equation + y = 0. +3- dx dx da (1) Write down the characteristic equation for (1). Factorize the third degree poly- nomial in this...
-
It is budgeting time again, and you would like to know the expected payoff (in terms of dollars collected) of spending an extra dollar on collection of delinquent accounts, after adjusting for the...
-
On July 1, 2021, Katrina purchased tax-exempt bonds (face value of $75,000) for $82,000. The bonds mature in five years, and the annual interest rate is 3%. a. How much interest income and/or...
-
The partially completed work sheet from the books of Lewis Music Store, a business owned by Hugo Lewis, for the year ended December 31, 20--, is shown on page 552. REQUIRED 1. Analyze the work sheet...
-
In general, matrix multiplication is not commutative (i.e., AB BA). However, in certain special cases the commutative property does hold. Show that: (a) If D1 and D2 are n n diagonal matrices, then...
-
Inflation (how much your money will buy) and interest rates (how much your money will earn if saved) are connectedand both of these economic factors should be considered in your financial planning,...
-
Runcke Motor Company manufactures automobiles. During September 2011 the company purchased 5,000 head lamps at a cost of $9 per lamp. Runcke withdrew 4,650 lamps from the warehouse during the month....
-
Morgan Company issues 20-year bonds with a face value of $20,000,000 and a stated annual interest rate of 5%. The bonds pay interest annually on December 31. The market rate of interest on the date...
-
Research the As , Bs , Cs , and Ds categories of Fintech players in today's digital era. Using that information: Define each player category, giving an example of a key company/institution in each...
-
Service shops are most likely to use: A.process costing B.product costing C.job costing D. hybrid costing
-
Which is of the highest value? (a) 1 bar (b) \(10^{5} \mathrm{~N} / \mathrm{m}^{2}\) (c) \(1 \mathrm{~atm}\) (d) \(100 \mathrm{kPa}\) (e) None of these
-
Aerofoils NACA3306 and NACA2412: which of them should have relatively higher lift coefficient?
-
Why is the coefficient lift of a cambered aerofoil " 0 "?
-
The Mach number of an airflow is \(M=0.7\). What is its speed coefficient \(M^{*}\) ?
-
What causes the significant rise in the drag coefficient as the freestream Mach number increases?
-
Question 13 U.S Telephone Cellular sells phones for $100. The unit variable cost per phone is $50 plus a selling commission of %10 (based on the unit sales price per phone). Fixed manufacturing costs...
-
An environmentalist wants to determine if the median amount of potassium (mg/L) in rainwater in Lincoln County, Nebraska, is different from that in the rainwater in Clarendon County, South Carolina....
-
If a nonlinear sys1em depends on a parameter le (such as a damping constant, spring constant, or chemical concentration), a critical value k0 where the qualitative behavior of the system changes is...
-
If the growth of a population follows the logistic model but is subject to "harvesting" (such as hunting or fishing), the model Becomes (a) Show that when the harvest rate h is constant, a maximum...
-
Base and Solution Spaces: For each of the differential equation in problems 1-3, give a basis and a solution space in terms of the basis? 1. y" - 4y' = 0 2. y" - 10y' + 25 y = 0 3. 5y" - 10y' - l5y =...
-
The time value of money Consider the following cash flows from today (year zero) to year 3 in pounds sterling: Your interest rate (or discount rate) in pounds is 10 percent. a Compute the net present...
-
Balance sheet hedging Gateaux Antoinette, SA, a subsidiary of Antoinette Cakes Inc., New York, had cash of 100,000 and accounts receivable of 100,000 each on its books (or 200,000 in total) when the...
-
The equilibrium exchange rate Consider the following quantity of dollars demanded and supplied daily in millions as a function of the dollars peso price, S: a. Solve for the equilibrium exchange rate...
Study smarter with the SolutionInn App