Let y1 and y2 be solutions of a second order homogeneous linear differential equation y'+ p(x) y' + q(x) y = 0 , in R. Suppose that y1(x) + y2(x) = e*, W[y:(x) , y2(x) ] = e* , where W [ y1, y2 ] is the Wronskian of y1 and y2 . a) Find p (x) b) Find q (x) c) Find the general form of yı and y2 .

Let y1 and y2 be solutions of a second order homogeneous linear differential equation y'+ p(x) y' + q(x) y = 0 , in R. Suppose that y1(x) + y2(x) = e, W[y:(x) , y2(x) ] = e , where W [ y1, y2 ] is the Wronskian of y1 and y2 . a) Find p (x) b) Find q (x) c) Find the general form of yı and y2 .

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Find the linear, nonhomogeneous differential equation with constant coefficients whose solution is y…

A: Given solution to a linear non-homogenous differential equation is yt=c1et+c2e2t-3tet+cost-e-t2…

Q: - It is known that y1 (x) = x and y2(x) = x are solutions of the differential equation xy" - 2xy +…

Q: Solve the partial differential equation t∂u/∂x−x∂u/∂t=0 with the initial condition u(x,0)=f(x),x>0.…

A: Hint: First compare the given partial differential equation with the general form quasilinear PDEs…

Q: Let be yı(x) and y2(x) solutions for a differential equation with constant lineal coefficients y" +…

Q: Let L(y) = anym)(x) + an – 1 y(n – 1)(x) + ·…· + a1 y'(x) + ao y(x) - where ao, a1, ,an are fixed…

A: Given: Ly1x=6xe8x, when y1x=36xe8x. Ly2x=5e8xsinx, when y2x=10e8xcosx. Ly3x=5e8xcosx, when…

Q: Let y1, Y2 be two independent solutions of (t – 1)y" – ty' + y = 0. If W (y1, Y2)(2) = e² , then…

Q: Consider g(x, y, z) = ey² + 2y - yArcsinz (a) Find the three (first-order) partial derivatives of g.…

A: Given that , gx,y,z=exyz+2xy-ysin-1z

Q: 25. Let y = y₁ (t) be a solution of y' + p(t)y = 0, and let y = y₂ (t) be a solution of y' + p(t)y =…

Q: Let y1 and y2 be solutions of a second order homogeneous linear differential equation y''+ p(x) y'…

A: p(X)= -1 q(X) = -2 Y(X)= c1e2x+c2e-x

Q: Let y1(x) and y2(x) be solutions of the linear differential equation with constant coefficients…

A: To find the possible solutions of given differential equation.

Q: Are fi (x) = e-3* and f2(x) = e solutions to the differential equation y" – y' – 12y = 0 on the…

Q: By assuming that u(x, y,t) has the product form u(x, y, t) = 4(x+ Vt)ø(x – Vt)Y9y), separate the…

Q: First, show that (0,0) is the only equilibrium of the differential equation (multiply x by x and y…

A: To find the equilibrium points, we set dx/dt and dy/dt to zero and solve for x and y…

Q: 5. Consider the nth-order homogeneous linear differential equation: y(m) +an-1y(n-1) '+….+a2y"…

Q: Use separation of variables to derive ordinary differential equations for the factors X(x) and Y(y)…

Q: (b) Use the method of separation of variables to determine the function u(x, y) which satisfies the…

A: Here given partial differential equation is ∂u∂x-∂u∂y=0 .......(1) and ux, 0=4e-3x

Q: 1. Consider the following partial differential equation: Bra(r, y) – 3(r, y) + 2u(x, y) = 0.(E). Let…

Q: A1 Compute the Wronskian of the functions y₁ = e, y₂ = 1 - e and use Theorem 1 to show that y₁, y2…

A: The Wronskian of two functions y₁ and y₂ is defined as:W(y₁, y₂) = y₁y₂' - y₂y₁'Therefore, for y₁ =…

Q: 2. Consider the first-order linear partial differential equation (2x+y) ux + 2y uy = x, x ≤ R, y >…

A: Step 1:Step 2: Step 3:

Q: Find all functions that form a fundamental set of solutions for t2y" – 2ty' + 2y = 0. O y(t) = et O…

A: We have to solve given problem:

Q: 3. (Sec. 3.5) Consider the nonhomogeneous second order linear DE Ly] =y" +p(t)y' + g(t)y=g(t), where…

Q: Find the linear, nonhomogeneous differential equation with constant coefficients whose solution is y…

A: We have to find homogeneous differential equation first.

Q: Consider applying the method of separation of variables with u (x,t) = X (x) T (t) to the partial…

Q: yı(t) = 4et, y',(t) = e=t ty" + 2ty' + ty = 0, a. Verify if the functions yı and y2 are solution of…

Q: Consider the equation y' – y – 2y = 0. Assume that y, (t) = e* and y, (1) = e2 form a fundamental…

Q: Consider applying the method of separation of variables with u(x, t) = X(x) T(t) to the partial…

Q: 1. A string has a partial differential equation with its initial conditions and boundar equation of…

Q: homogeneous equation

Q: a) The functions f(x) = e¬x, g(x) = cosh x, and h(x) = sinh x are solutions of the differ- ential…

A: 1.a) Given fx=e-x, gx=coshx, hx=sinhx. To check if they are linearly independent using Wronskian.

Concept explainers

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

Let y, and y2 be solutions of a second order homogeneous linear differential
equation y"+ p(x) y' + q(x) y = 0 , in R. Suppose that y1(x) + y2(x) = e* ,
W[ y (x) , y2(x) ] = e* , where W [y1 , y2 ] is the Wronskian of yı and y2 .
a) Find p (x)
b) Find q (x)
c) Find the general form of yı and y2 .

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Trending now

This is a popular solution!

Step by step

Solved in 5 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

2. Suppose that y₁ is a solution for of the homogeneous SOLDE " + p(t)y' + q(t)y = 0. (a) Explain how to find a non constant function tv(t) such that y2(t) = v(t)y₁(t) is also a solution. (b) Explain why y₁ and y₂ are linearly independent.
A. Classify the given DE according to order, degree and type Differential Equation Order Degree Туре 1. (2xy + x²)dx + 2ydy = 0 d³y, 2. ()3 - x()² + x² = 0 'dx³- 'dx²' 3. (x - 1)²y" + xy' + y = 0 əx 'Xze' 4. + 5y(2 = 8 ду ду? d³y 5. dx³ x2 - 3x = 4
2. First verify whether y(x) satisfy the given Differential Equation. Then determine C dy a) х— a) x+3y-2x';y(x) =x' + Cx,y(2) = 1 dx b) y'+y tanx = cosx, y(x) = (x+c)cosx, y(x) = 0 c) y' = 2y, y(x)= ce* - 1, y(0) = 5
EXERCISE 1C Legendre's differential equation is given - (1-x²)y" — 2xy' + a(a + 1)y = 0, y = y(x),a ≤R, (1). Show that the general solution of (1) is y = aoy₁ (x) + a₁₂3₂ (x), where and y₁(x) = 1 a(a − 1) 2 +00 x² - Σ² n2=2 a(a + 1)/ 2n a(a +1D).... 1. 6 Y₂(x) = x + Σ₁¹ (1 — ala+¹))... (1 +00 1 1 - 2n+1 2 a(a + 1) (2n-2) (2n-1)' a(a+1) 2n(2n-1) -2n+1 --)x² -)x²n 2n
Consider the following partial differential equation: y(e +1)u, – e"(y² + 1)²u, = 0, y >0 1. By using the separation method with u(r, y) = f(r)+ g(y) to solve (F), we obtain that : (F) 1+ e" df (y² + 1)² dg a) da fip df b) 1+ er dr (y? + 1)? dg dy dg et %3D 1+ e" df c) et dr (y? +1)2 dy d) None of the above. 2. A solution u(x, y) of (F) is ( B is an arbitrary constant): a) u(r, y) = À (In(1+ e*) – In(y² +1)) + B. 1 b) u(x, y) = (In(1 + e")- + B. 2(y² + 1) 1 c) u(r, y) = A + B. 2(y? + 1). d) None af the above.
R.solanki
3.) Given that y, (t) = (t + 2)- e' and y,(t) = e' - 2 are both solutions of a certain second order homogeneous linear differential equation: y" + Ay' + By = 0 Answer each question below. For each true/false question, state a brief reason that justifies your answer. a.) Show that these two solutions are linearly independent. b.) True or False: y, and y, form a fundamental set of solutions of this equation. c.) True or False: y,(t) = (t + 3) · e – 2 is also a solution of the equation d.) True or False: y,(e) = (t + 1) - e' is also a solution of the equation
General integral of the partial differential equation, (y + zx)=, -(x+ yz)=, =x² – y² is, y+:)= 0 (b) F(x² +y² -:²,xy+:)=0 (c) F(x² – y² - =²,xy +:)= 0 (d) F(x² + y² +z²,xy- :)=0 (a) F(x² + y² +=²,.
4. Are y1 (x) = e*and y2 (x) = x +1 solutions of the D.E. xy" – (x + 1)y' + y = x2. Are y1(x) independent? and y2(x) linearly
Examine the linearity states, orders and degrees of the following differential equations.