Particular solution Definition and 104 Threads

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ordinary is used in contrast with the term partial differential equation which may be with respect to more than one independent variable.

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  1. T

    A particular solution to the differential equation y'' + 2y' + y = t^2 + 3?

    A particular solution to the differential equation y'' + 2y' + y = t^2 + 3? is t^2 - 4t +7 in the answers, but i get t^2 - 4t + 9 so where am i going wrong... y = Ay^2 + by + c y' = 2Ay + B y'' = 2A 2A + 2(2Ay + B) + Ay^2 +By + c = t^2 + 3 A(y^2) + (4A+B)y +(2A +2B+c) = t^2 + 0t + 3...
  2. R

    Finding a Particular Solution for a Non-Homogeneous PDE

    Homework Statement Find the solution of yu_x + xu_y = (y-x)e^{x-y} that satisfies the auxiliary condition u(x,0) = x^4 + e^x Homework Equations Given in question The Attempt at a Solution The general solution to this is u(x,y) = f(y^2-x^2) Applying the auxiliary condition I...
  3. C

    Trouble with particular solution of differential equation - rewrite my answer?

    Trouble with particular solution of differential equation - [SOLVED] Homework Statement Find the particular solution of the differential equation satisfying the initial condition The Attempt at a Solution I end up with (1/2)ln(-x^2+10x) which does satisfy the initial conditions...
  4. T

    Sum of particular solution and homogenous solution of differential equation

    Homework Statement The Attempt at a Solution Suppose there is a solution X^S ≠ X^p + X^c. Let Y_1,...,Y_n be the set of linearly independent solutions whose span gives the general solution to the differential equation. Let us write the general solution to the differential equation as a...
  5. P

    Particular solution to 4th order ode

    hey, i have this 4th order ode question that I've been working on, the homogeneous solution was easy enough by finding the particular solution has become a bit annoying, the ode is y'''' - 4y'' = 5x2 - e2x I have gotten the particular solution using variation of parameters...
  6. H

    Find particular solution third order Diff Eq

    Homework Statement y''' - y = e^x + 7 Homework Equations The Attempt at a Solution I used y=Ae^x +B and then I multiplied by x^2 because y_c = c1 + c2 e^x + c3 e^(-x) the c1 and c2 e^x value repeat. Therefore I got: y= Ax^2 e^x + Bx^2 I got A = 0 and A=1 which is wrong...
  7. K

    Obtaining particular solution of second order linear DE from first

    In cullen-zill chapter 6 equation 23 it says that y_{2}(x)=y_{1}(x)\int\frac{e^{-\int P(x)dx}}{y_{1}^{2}(x)}dx is a solution of y''+P(x)y'+Q(x)y=0 whenever y_{1}(x) is a known solution Where does this come from? I would like to be able to prove this or find a proof somewhere. My...
  8. T

    Find the particular solution of a differential equation

    Homework Statement Find the particular solution of the differential equation dy/dx = (x-4)e^(-2y) Satisfying initial condition y(4)=ln(4) Homework Equations N/A The Attempt at a Solution I separated this into dy/e^(-2y) = (x-4)dx I then integrated it to get e^(2y)/2 = x^2/2 - 4x I then...
  9. A

    Find a particular solution to the differential equation?

    Homework Statement Find a particular solution to the differential equation -6y''+5y'-1y = -1t^2+1t+1e^2t Homework Equations y = y(c) + y(p) The Attempt at a Solution First solve the homogeneous equation -6y" + 5y' - y = 0. The characteristic equation is -6r^2 + 5r - 1 = 0...
  10. T

    Particular Solution of ODE using Annihilator

    Homework Statement By using the method of differential operators, solve y''+2y'+2y=2e-xsinx 1. Determine what is the annihilator of the inhomogeneous term. 2. Find a particular solution. 3. Write the general solution for the equation. Homework Equations xneaxsin(bx) --> annihilated by...
  11. P

    Finding particular solution yp of given equation

    y''+2y'-3y=1+xex ive tried yp=Axex+B and yp=Ax2ex+Bx and both don't work. assuming I am doing it correctly. im given the answer as: 1/3+1/16(2x2-x)ex suggestions? thanks
  12. C

    Non homogeneous ODE particular solution using power series

    Homework Statement Find the particular solution to the ODE y"+y=x using power series Homework Equations y=\sum(a_{n}x^{n})The Attempt at a Solution i tried plugging in y=\sum(a_{n}x^{n}) into the original equation and comparing coefficients of x to the first degree, but i am not sure how to...
  13. I

    Help with finding particular solution of a 2nd order linear ode

    Homework Statement obtain the general solution y(x) of y''-2y'+y=e^(2x)/(e^x+1)^2 Homework Equations variation of parameters The Attempt at a Solution I have obtained the continuous equation. I tried two methods of variation of parameters, but both of them got me stuck. 1...
  14. P

    Solve Particular Solution Differential Equation: {\theta}''(t)-{\theta}(t)=tsint

    Homework Statement Find a particular solution to the differential equation: {\theta}''(t)-{\theta}(t)=tsint The attempt at a solution So I started by using the particular solution {\theta}_{p}=(At+B)(Csint+Dcost) Before I continue with the rest of the solution, is this correct so far?
  15. S

    How to Solve Nonhomogeneous Euler Equations with Absolute Values?

    Homework Statement Solve the IVP (x^2)y'' + 4xy' - 40y = x^6 for y(1) = 10, y'(1) = 1Homework Equations not so much "equations" but here I try to use variation of parameters to get the particular solution.The Attempt at a Solution FOR THE HOMOGENEOUS SOLUTION: using the substitution y = x^r...
  16. T

    Particular Sub y_p into DE for r(x)

    Homework Statement Show (by substituting it directly into the differential equation) that \displaystyle y_p = y_2 \int \frac{ry_1}{W}\;dx - y_1 \int \frac{ry_2}{W}\;dx is a particular solution of y'' + p(x)y' + q(x)y = r(x). Homework Equations W is the Wronskian y_1 y_2^{\prime}...
  17. Z

    2nd order DE: finding particular solution

    Homework Statement Find the general solution of the following differential equation: y" + 3y' + 2y = sin ex Homework Equations y = yh + yp homogeneous solution: (found by solving characteristic eq) yh = Ae-2x + Be^-x The Attempt at a Solution from my table if r(x) = ksin(wx)...
  18. J

    Find the particular solution of this difference equation (solve most of it)

    Homework Statement Find the particular solution. y(k+2) + y(k+1) -6y(k) = 3^(k) Homework Equations The Attempt at a Solution Still need to be answered Question (1): Before finding the particular solution, is it true that we should ALWAYS get the homogeneous solution...
  19. 0

    Particular Solution of an Inhomogeneous Second Order ODE

    Homework Statement A particular solution of y'' + 4y = tanx Answer choices are: (a) 1/2*cos(2x)ln|sec(2x)+tan(2x)| (b) -1/2*cos(2x)ln|sec(2x)+tan(2x)| (c) 1/2*sin(2x)(ln*cos(x)+x*sec(2x)) (d) 1/2*sin(2x)(ln*cos(x)-x*sec(2x)) (e) none of the above Homework Equations The...
  20. D

    Fourier series for particular solution when n=1

    hi, i already got the Fourier series for f(x) = x where -pi/2 =< x =< pi/2 which is f(x) = sigma, n=1 to infinity ( (-1)^n+1*sin (2nx) / n ) in order to find particular solution for y'' + 4y = f(x) i have to equate with with y(x)_p = A0 + sigma, n=1 to infinity (An*cos(2nx) + Bn*sin(2nx))...
  21. jegues

    Form of a particular solution for N.H.L.D.E. w/ constant coefficients

    Homework Statement You are given that the roots of the auxiliary equation associated with the linear, differential equation \phi(D)y = 2x- 3xe^{-3x} are m = \pm2,0,0. Write down the form of a particular solution of the differential equation as predicted by the method of undetermined...
  22. E

    Finding Particular Steps of Derivation

    Find the Particular Solution. Show the steps of derivation, beginning w/ the general solution. y'=y/x+(2x3/y)cos(x2),y(sqr(pi/2))=sqr(pi) legend: sqr=square root pi= pi.. the 3.1416 i nid the steps.. thanks... :)
  23. P

    Solve Electrical Circuit Problem: Particular Solution

    Hi everybody. I've got kind of a problem solving the following problem, so really hope for some help. The task says: -------------------------- The figure beneath shows an electrical circuit containing following components: a resistance R, a capacitor with the capacitance C and finally a...
  24. L

    Particular Solution to a System of First Order Linear Equations

    Consider a certain linear system of 2 equations x'=Ax. Suppose the real-valued coefficient matrix A has r=-1+4i as one of its eigenvalues, and that one of its corresponding eigenvectors is (2, 5-2i). Find the particular solution that satisfies the initial condition x(0) = (-4, 3)...
  25. N

    Determining particular solution with matlab

    Homework Statement 4 -2 0 -2 | 32 -2 4 -2 0 | 0 0 -2 4 -2 | 0 -2 0 -2 4 | -32So I am trying to have MATLAB come up with the answer for this with a,b,c,d as arbitrary. Homework Equations The Attempt at a Solution I have tried...
  26. A

    Particular Solution to a Second Order Ordinary Differential Equation

    Homework Statement Find the Particular Solution to the differential Equation satisfying the initial conditions. y''+7y'+10y= -30 y(0)= 3 y'(0) = -27Homework Equations Characteristic Equation for Homogenous Solution y''+7y'+10y=0 roots are -2 and -5 General Solution is C1e^-2t + C2e^-5t...
  27. G

    ODE - Proof of a Particular Solution

    Homework Statement Homework Equations (1) y\prime\prime+p(x)y\prime+q(x)y=g(x) (2) y\prime\prime+p(x)y\prime+q(x)y=0 If y1 and y2 are complimentary solutions of (2), then a particular solution of (1) is given by Yp = u1*y1 + u2*y2 The Attempt at a Solution Anyone have a good...
  28. F

    Form of particular solution for y''-2y'+y=(e^2)/x

    Homework Statement Find the general solution for: y''-2y'+y=ex/x Homework Equations NONE - not an initial value problemThe Attempt at a Solution Solve the homogeneous first: r2-2r+1=0 r=1 as a double root So: y1=c1ex y2=c2xex ...but what in God's name is the form for the particular Y (based...
  29. Q

    Why is there a sin term in the particular solution for cosine equation?

    Homework Statement Q'' + 100Q' +50000Q = 4000cos(100t) i found the general solution to be e-50t[Acos(50sqrt19)t +Bsin(50sqrt19)t] but i have a problem with the particular solution i tried Cei100t did i try the wrong expression? because when i compared coefficients, i found...
  30. S

    Find the particular solution of the differential equation

    Find the particular solution of the differential equation dy/dx = (3x+42y)/7x satisfying the initial condition y(1) = 5. Attempt: dy/dx = 3/7 + 6y/x dy/dx - 6y/x = 3/7 p(x) = -6/x q(x) = 3/7 u(x) = -6 ∫(1/x)dx = -6ln|x| e^(u(x)) = -x^6 1/e^(u(x)) ∫e^(u(x))q(x)dx =...
  31. S

    Solving Inhomogeneous Differential Equations with ODE Particular Solution

    Hi all, I've been staring at this for much to long: y''+2y=6x+4 what were going over now is homogeneous and inhomogeneous differential equations but i can't seem to think of a method for solving this one other then by inspection. Thank you very much for your help.
  32. K

    Find a particular solution that satisfies the intial condition

    Homework Statement 2xy'-ln x2=0 y(1)=2 Homework Equations The Attempt at a Solution 2x(dy/dx)-ln x2=0 I think I'm suppose to separate variables and then integrate next but I'm not sure.
  33. J

    Finding a particular solution of a differential equation

    Homework Statement Find the particular solution of this differential equation: y`` −3y` −10y=10t2+16t−19 Homework Equations The Attempt at a Solution I'm not really sure what the roots look like for 10t^2 + 16t - 19. I thought t had roots (0,0). Does that mean t^2 has roots...
  34. M

    Solving a Differential Equation: Finding the Particular Solution

    I have to find the particular solution to the differential equation: (-21/4)y''+2y'+y=4xe^(3x) First, I chose my trial function to be yp=(Ax+B)*e^(3x). Is this correct? so yp'=3(Ax+B)*e^(3x) yp''=9(Ax+B)*e^(3x) So I plug these into the differential equation and I get...
  35. M

    Complexification for finding a particular solution

    So I have this: y'' + 4 y = 4 sec(2 t). which translates to p(D)y=4sec(2t) where p(D)=D^2+4 where D is a differential operatior I know i have two choices for this, which is either looking for the particular solution through variable parameters which involved the winkonsian and some...
  36. J

    Finding the particular solution to an ODE with set boundary conditions.

    Homework Statement Homework Equations N/A The Attempt at a Solution The problem and attempt are as above, I'm not sure where to go from here though. I'm not sure what to do with the boundary condition of dx/dt=-2 and t=0. Any help appreciated.
  37. N

    Finding particular solution with fourier series

    I was wondering what to do if I have a DE like this x'' + \omega^2x = \cos^2(t) \sin^2(t) I have to decide for what omega it has a solution with period 2pi. Now to solve this I have to find the Fourier series representation of the right hand side, but the problem is that I get all An...
  38. S

    Mechanics help,find the particular solution and amplitude

    a block P of mass m lies on a smooth plane AB that is inclined at angle (alpha) to the horizontal.The block is attatched to the bottom of the plane,A, by a spring of stiffness 2k and natural length L0.The block is initially released from rest from the equilibrium position.the equilibrium...
  39. S

    Particular solution / undetermined coefficients

    Homework Statement tyʺ+yʹ=4t Homework Equations The Attempt at a Solution The problem that I am having with this problem is I've never been shown how to calculate the particular solution when there is an unknown (t) on the left hand side of the equation. If the problem were...
  40. W

    2nd order differential - particular solution

    [SOLVED] 2nd order differential - particular solution Homework Statement a) Find the general solution of the differential equation: \[2\frac{{d^2 x}}{{dt^2 }} + 5\frac{{dx}}{{dt}} + 2x = 2t + 9\] b) Find the particular solution of this differential equation for which: \[x =...
  41. F

    Getting the particular solution of this differential equation.

    Homework Statement y''-2y'-3y=-3t*e^(-t) Homework Equations Has to be done with method of undetermined coefficients The Attempt at a Solution the chacteristic equation is: c1*e^(3t) + c2*e^(-t) my attempt at Yp is (a*t+b)*e^(-t)... so you that's not it. i tried many versions...
  42. E

    Finding a particular solution (Diff. Eq.)

    Homework Statement I seem to not fully understand how to find particular solutions. I'm having a hard time guessing what the solutions maybe. I'll explain as I write out the problem. y''' - 2y'' + y' = te^t + t^2 + 4, find the general solution. Homework Equations Z(r) = r^3 - 2r^2 +...
  43. D

    ODE particular solution of the homogeneous equation

    y''(t)+A^2y(t)=f(t), t>0, y(0)=B, y'(0)=C, A, B, C\in\mathbb{R} e^{iAt} is a particular solution of the homogeneous equation. I can multiply it by some arbitrary function and find another solution of the homogeneous case, but when I try with the f(t) on the RHS, I can't do it. Anyone help?
  44. B

    Finding Particular Solution for System of Equations

    Hi, I'm having trouble finding the particular solution of the following system. \left[ {\begin{array}{*{20}c} {\mathop x\limits^ \bullet } \\ {\mathop y\limits^ \bullet } \\ \end{array}} \right] = \left[ {\begin{array}{*{20}c} 1 & { - 1} \\ { - 1} & 1 \\ \end{array}}...
  45. D

    ODEs: Why do I need Both a General And a Particular Solution?

    I am asking this question as it relates to physics, and in particular how it relates to harmonic oscillation. Why is the equation not solved when I use only a particular solution? Why is the equation not solved when I use only a general solution?
  46. O

    Finding the particular solution of DE

    I have a nonhomogeneous DE and wants to find the particular solution for Asin(x)sin(t) Is there any tips in using method of undetermined coefficient to guess the particular solution of this?
  47. K

    O.D.E. Complementary and Particular Solution

    Not exactly homework, but it is a problem I'm having... Im given an ode that reads: y"-2y'-3y = 6; y_c = C_1 * /exp^-x + C_2 * /exp^3x y_p is -2 y(0) = 3 y'(0) = 11 Now I am tasked to find what C_1 and C_2 are. I know that y(x) = y_c + y_p so: y = C_1 * /exp^-x + C_2 * /exp^3x...
  48. U

    Dif eq with particular solution

    find the particular solution for y''+25y=50sin(5t) I am using variation of parameter: y_p=U_1e^{5x}+U_2e^{-5x} y_1=e^{5x},y_1'=5e^{5x},y_2=e^{-5x},y_2'=-5e^{-5x} U_1=- \int \frac{e^{-5x}50sin(5t)}{-10}=-0.5e^{-5x} (cos5x+sin5x) U_2= \int \frac{e^{5x}50sin(5t)}{-10}=0.5e^{5x}...
  49. U

    Particular solution for dif eq

    -3y''-2y'+y=-t^2+2t+2e^{-4t} i am to find the particular solution to this. i started with the non-exponential: y=At^2+Bt y'=2At+B y''=2A (-6A-2B)+(-4A+B)t+A(t^2) -6A-2B=0,-4A+B=2,A(t^2)=-1 A=-1, B=3 i started with the exponential: y=Ce^{-4y} y'=-4Ce^{-4t} y''=16Ce^{-4t}...
  50. E

    Driven, damped harmonic oscillator - with particular solution

    Driven, damped harmonic oscillator -- need help with particular solution Consider a damped oscillator with Beta = w/4 driven by F=A1cos(wt)+A2cos(3wt). Find x(t). I know that x(t) is the solution to the system with the above drive force. I know that if an external driving force applied...
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