What is Initial value problem: Definition and 178 Discussions

In multivariable calculus, an initial value problem[a] (ivp) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem.

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  1. C

    Question on initial value problem

    I am working on the following initial value problem: y'' + 4y = 4, y(0) = 1, y'(0) = 1 The method they show is: (1): y = A cos x + B sin x A = 1, (2): y = 1 + A cos x + B sin x y(0) = 1 + A = 0 y'(0) = B = 1 Final answer is y(x) = 1 + sin(x) The problem I don't see is the first...
  2. C

    Solving the Initial value problem

    Homework Statement dx/ dt = x + y dy/ dt = x + y + et x(0) = 0 y(0) = 1.Homework Equations The Attempt at a Solutionx'' = x' + y' = x' + x + y + et x'' - 2x' = 0 y = x' - x
  3. T

    Solve the Initial Value Problem

    Homework Statement y' = (2x) / (y+(x^2)y) y(0) = -2 The Attempt at a Solution I tried doing this by finding the Integrating factor and I got that to be u = -1-x^2 by using the (My - Nx) / N formula. Using this did not work out for me and I'm not seeing the other approach...
  4. A

    Finding the Constants in a Second Order Differential Equation

    Homework Statement d2y/dx2 = 2-6x Given: y(0)=-3 and y'(0)=4 Homework Equations None that I know of. The Attempt at a Solution I know that for a single order derivative you would just find the integral, set y=1 and x=0. But I'm confused because here we're given two...
  5. B

    Initial Value Problem. Laplace Transform

    Homework Statement The question is solve using laplace transforms: y" - 4y = 1-2t where y(0) =0 and y'(0) = 0 Ans is: y = t/2 - 1/4 + 1/4*e^(-2t) Homework Equations http://www.stanford.edu/~boyd/ee102/laplace-table.pdf" partial fractionsThe Attempt at a Solution (s^2)Y - sy(0) - y'(0) - 4Y =...
  6. T

    Initial Value Problem, confused due to non-linearity

    Homework Statement The equation 2y'' - y' + y^2(1-y) = 0 is a special case of an equation used as a model for nerve conduction, and describes the shape of a wave of electrical activity transmitted along a nerve fibre. Find a value of the constant a so that y = (1 + e^(ax))^(-1) is a...
  7. H

    Solving Initial Value Problems Using Integration | f'(x)=(x^2-1)/x^5, f(1/2)=3

    Homework Statement Solve each of the following initial value problems: a) f'(x)=(x2-1)/x5 f(1/2)=3 Homework Equations The Attempt at a Solution I guess my problem with this is I'm not 100% sure what I'm being asked to do. I know I need to use integration to find the original...
  8. jegues

    Solve the initial value problem

    Homework Statement Solve the following initial value problem y^{''} + 2y^{'} - 15y = 4\delta(t-2), \quad y(0) = 1, \quad y'(0) = -1. Homework Equations The Attempt at a Solution See figure attached for my attempt at the solution. Does anyone see any problems? Sorry if...
  9. G

    Initial value problem for system of two diff. equation

    Hi all! I have quiet a lot a problems solving and programing with Mathematica this system of diff. equations: y'[x] == 1/b[x], b'[x] == 1/ubb 1/(y[x] + R1) 1/b[x] (uba 1/(y[x] + R1) - ub), b[0] == b00, y[0] == 0 y[2] == Sqrt[5] where ubb, uba, ub are functions: ub = b[x] -...
  10. D

    Solution to an initial value problem

    Homework Statement dy/dx=5y(1-y) y(0)=0.1 x belongs to [0; 2] Homework Equations The Attempt at a Solution dy/5y(1-y)=dx using partial fractions (1/(-y^2+y))=1/(y(1-y))= A/y+B/1-y multiplying both sides by (y(1-y)) gives 1=A(1-y)+B(y) when y=0 1=A so A=1 when y=1...
  11. M

    Initial value problem - value returns invalid operation

    Homework Statement Solve the initial value problem Given: x=0, y=-2Homework Equations 2x(y+1)dx - ydy = 0The Attempt at a Solution Manipulating the given equation: --> 2x(y+1)dx - ydy = 0 --> 2xdx = ydy/(y+1) --> 2xdx = (1 - 1/y+1)dy --> 2x2/2 = y - ln(y+1) + lnC (where lnC is a constant)...
  12. A

    What is the solution to solving a linear differential equation?

    Homework Statement 1. To use Euler midpoint method to find an approx value to y(0.1) using a step size 0.1 2. To use integrating factor method to find exact value of y(0.1) 3. Determine the global error ( min 5 decimal places) Homework Equations f(x,y) = x+y, y(0) = 1.35 The...
  13. H

    Differential Equations initial value problem

    Homework Statement Let f(t) be the solution to the initial value problem 2t(dy/dt)+y=t^4 with f(0) = 0 find f(t). Homework Equations The Attempt at a Solution I tried to do this by separating variables but that hasn't gotten me very far. I don't know if I can do it by doing...
  14. J

    Solving Initial Value Problem with Implicit Solution

    Homework Statement Solve the initial value problem (y+e^-y)y'=sinx subject to y(pi)=0 Homework Equations The Attempt at a Solution I'm not quite sure what to do with this one. I've scanned through my book and could find no similar problems in what we've done so far. I tried to...
  15. F

    Solve the initial value problem?

    1) (dy/dx) = (1) / ( 2√x ) + 10 , y(1) = -1 2) (d^(2)r / dt^2) = ( 4 / t^3 ) ; (dr/dt) | t=1 = -5, r(1)=5 ... i have no idea where to start. what are the steps needed to figure out the answer...can someone please solve this for me! t thank you so much.
  16. P

    Initial Value Problem with Laplace Transforms

    Solve the following given y(0) = 0 & y'(0)=1: y′′+3y′+2y = u2(t), such that u2(t) is a heaviside step function Here's what I've got so far, =>s2Y(s)−sy(0)−y′(0) + 3sY(s)−3y(0) + 2Y(s)= exp(−2s)/s Y(s) = (exp(−2s) + s) / (s(s2+3s+2)) Y(s) = exp(−2s)/(s(s2+3s+2))* + 1/(s2+3s+2)** The...
  17. D

    Please check my work on initial value problem. thank you

    Homework Statement Solve the following initial value problem. Sketch the solution and describe its behavior as t increases. y'' + 4y' + 3y = 0 y(0) = 2 y'(0) = -1 1st i solved the characteristic: r^2 + 4r + 3 = 0 r = -1 r = -3 then the general solution is; y = c_1e^-x +...
  18. N

    Differential Eq. Initial Value Problem (separable/partial fractions/hyperbolic)

    Homework Statement dx/dt = 2*(1-x^2) with x(0)=0 Obtain the analytic solution to the IVP. Then, obtain the terms in the Taylor series up to and including the t^5 term. Note: should be possible to express x(t) in terms of hyperbolic functions. The Attempt at a Solution...
  19. J

    Laplace transform initial value problem

    Homework Statement Use the Laplace transform to solve the following initial value problem: x' = 7 x + 5 y, y'= -2 x + e5t, x(0)=0, y(0)=0 Find the expressions you obtain by taking the Laplace transform of both differential equations and solving for Y(s) and X(s) Homework Equations...
  20. D

    General solution of initial value problem -dont understand problem is asking me?

    General solution of initial value problem --dont understand problem is asking me?? Homework Statement Find a value for y-sub-0 for which the solution of the initial value problem: y' - y = 1+ 3sin t y(0) - y-sub-0 remains finite as t approaches infinity. (i called it "y-sub-0" , just...
  21. M

    Is Boundedness the Key to Proving Uniqueness in Initial Value Problems?

    Hi all! I'm having some trouble with the Existence and Uniqueness (E&U) thms. for ODEs. Consider the IVP: x'(t)=f(t,x(t)), x(\xi)=\eta In order to prove Existence, we need f to be Lipschitz or at least locally Lipschitz. For the Uniqueness we use the Banach's fixed point thm. for the...
  22. E

    Solving Second Order Differential Equations for Beginners

    Homework Statement How can I find a general solution to this problem: 5y" + 9y' - 9y = 0 Homework Equations The Attempt at a Solution I am confused with second order different equations in it
  23. M

    Differential Equations - Initial Value Problem

    Homework Statement Suppose that the initial value problem y' = 7(x^2) + (5y^2) − 6, y(0)=−2 has a solution in an interval about x=0. Find y'(0) = Find y''(0) = Find y'''(0) = Homework Equations get it into standard form: dy/dt + p(t)y = g(t) find integrating factor =...
  24. E

    Solve Initial Value Problem Using Exponential Functions | Step-by-Step Guide"

    Homework Statement http://img40.imageshack.us/img40/7032/48975587.th.jpg http://img42.imageshack.us/img42/7264/solve2.jpg Homework Equations The Attempt at a Solution what I get is m^3e^(mx) + 10m^2e^(mx) + 9me^(mx) = 0 e^(mx) * (m^2 + 10m + 9) e^(mx) * (m+9)(m+1) m = -9 and m = -1 in...
  25. W

    Solving Initial Value Problem with f(x)=1 and erf(x)

    Homework Statement Consider the initial value problem y'+e^(x)y=f(x), y(0)=1.Express the solution of the initial-value problem for x>0 as a non elementary integral when f(x)=1 and also in term of erf(x) Can somebody please help me solve this question? Thanks!
  26. M

    Determination of Unknown Constants in Differential Equations

    What is the general rule for determining how many initial conditions one needs to uniquely determine a solution of a set of ODEs or PDEs? is it simply the number of derivatives? How about for coupled differential equations? cheers M
  27. J

    Second order initial value problem

    Homework Statement Solve the initial value problem y" = 2x - y' , y'(0) = 1 , y(0) = 0 I know this is probably a simple problem but I don't have a book for the class yet and the teacher didn't really cover this material in class but we still have homework due on monday so i need to figure...
  28. B

    Answer check: differential initial value problem

    Homework Statement solve the IVP y''-8y'+16y=0 y(0)=2 y'(0)=7 Homework Equations The Attempt at a Solution auxiliary equation: r2-8r+16=0 (r-4)2=0 so we have r=4 with m=2 y(x)= c1e4x+c2xe4x y(0)--> c1e0 + 0 = 2 c1=2 now we have: y(x)=2e4x+c2xe4x take...
  29. T

    MacLaurin Series solution to initial value problem

    Homework Statement Find the first six nonvanishing terms in the Maclaurin series solution of the initial value problem (x^2 - 3)y''(x) + 2xy'(x) = 0 where y(0) = y0 and y'(0) = y1. Homework Equations The Attempt at a Solution Should with just something like Φ(x) such that Φ(x) =...
  30. C

    Differential Equation - Initial Value Problem

    Homework Statement Solve the following IVP: \frac{dy}{dt} = \frac{1}{2y+3} y(0) = 1 Homework Equations The Attempt at a Solution \frac{dy}{dt} = \frac{1}{2y+3} \Rightarrow \int 2y + 3 dy= \int dt \Rightarrow y^2 + 3y= t +c where C is a constant Kinda stuck now. Any hints?
  31. M

    How did this second-order initial value problem do this?

    Homework Statement Find a second-order initial-value problem whose solution is sin at The Attempt at a Solution so i know that y' = a cos at y" = -a^2 sin at but the solution says that "Then y(t) = sin at is a solution to 2 y ''+ a y = 0 . Our initial conditions must be...
  32. L

    Laplace initial value problem> HELP PLEASE

    Laplace initial value problem... HELP! PLEASE! -------------------------------------------------------------------------------- Hello all! I'm stuck on this question: y''(t)+ 4y'(t) = sin2t y(0) = 0 solve it using laplace transform,... my final is tomorrow, and its 2 am, i...
  33. D

    Differential equations - solving initial value problem

    Homework Statement Solve the given initial value problem and determine how the interval in which the solution exists depends on the initial value y0. y' = -4t/y, y(0) = y0 Homework Equations The Attempt at a Solution I solved the DE and got to: y = +/- sqrt(C - 4t^2)...
  34. B

    Help with initial value problem

    Solve: dy/dt = 4y-2, y(0)=3 I first separate variables: dy/(4y-2) = dt I then integrate both sides to get: ln|4y-2|=t + c I next set 4y-2=Ce^t Do I just plug 3 in for y to get C=10 or am I going about this problem the wrong way??
  35. B

    Help with a simple initial value problem

    Homework Statement Solve the IVP dy/dt=4y-2, y(0)=3 This is how I work it. I integrate both sides and get y= 2y^2 - 2y + C I then solve for C: 3=2(0)^2 - 2(0) + C C = 3 Is this the correct way to solve for C?
  36. B

    Initial value problem D.E. (confused of a weird sol.)

    Homework Statement solve: y" + 4y' + 4y = (3 + x)e-2x ... (1); y(0) = 2 & y'(0) = 5 Homework Equations using Undetermined Coefficients Method The Attempt at a Solution solving the associated homog. eq of (1) : y" + 4y' + 4y = 0 ... (2) we get: yc = c1 e-2x as m=-2 let...
  37. Q

    Laplace transformation initial value problem

    Homework Statement I need to use the Laplace transformations to solve this IVP y'-y=2cost(5t) , y(0)=0 Homework Equations The Attempt at a Solution I separated the question until the point Y(s)=(2s)/(s^2+25)(s+1) and then 2s=A(s+1)+B(s^2+25) I get the...
  38. A

    Does the Existence Theorem Contradict the Initial Value Problem?

    Homework Statement show the initial value problem x(dy/dx)=4y, y(0)=1 has no solution. does this contradict the existence theorem. please explain The Attempt at a Solution it is easy to find out a general solution is y=C*e^(4x), C is a constant. and for any x the right part of the...
  39. S

    HELP Initial Value Problem Question-Differential Equations and Euler's Method

    URGENT HELP! Initial Value Problem Question-Differential Equations and Euler's Method Homework Statement This is just an extension of an earlier thread. I see now that they want me to use Euler's method so it might change the way I do the problem. The problem wants me to solve the initial...
  40. S

    Initial Value Problem Question - Differential Equations

    Homework Statement Solve the initial value problem where y(0) = 2.Homework Equations dy/dt = 3 - 5(y^(1/2))The Attempt at a Solution I tried the separable equation method but when it came time to take the integral of 1/[3 - 5(y^(1/2)], every solution I got became too complex to solve for y...
  41. A

    Exp Matrix to solve an initial value problem

    Exp Matrix to solve an initial value problem(urgent) Homework Statement Given the matrix A = \left[\[\begin{array}{ccc} 2 & 2 \\ 1& 3 \end{array}\right] a) Find the e^{tA} 2) Solve the x' = Ax + (1,0) \begin{array}{c} \end{array} where x(0) = (0,0) The Attempt at a Solution a)...
  42. 2

    Solve Initial Value Problem y'=y^2: Solution & Interval Determination

    Homework Statement Solve the initial value problem y'=y^2 , y(0)=1 and determine the interval in which the solution exists. The Attempt at a Solution So after solving the differential equation, we get y= 1/(1-t) However, I don't understand why the interval is only (- \infty , 1 )...
  43. M

    Can anyone check my work? 1st Order ODE Initial Value Problem (Repost)

    Repost with attachment >< 1. Homework Statement dy/dt + ty/(1+t^2) = t/(1+t^2)^1/2 y(1) = 2 Need to solve this initial value problem. The equation is a 1st order ODE 2. Homework Equations 3. The Attempt at a Solution I've attached my solution to the problem. Just...
  44. M

    Can anyone check my work? 1st Order ODE Initial Value Problem

    Homework Statement dy/dt + ty/(1+t^2) = t/(1+t^2)^1/2 y(1) = 2 Need to solve this initial value problem. The equation is a 1st order ODE Homework Equations The Attempt at a Solution I've attached my solution to the problem. Just wondering if anyone can check my work. I...
  45. S

    Differential in a initial value problem

    I am trying to solve a differential equation using an integrating factor and the term I am trying to differentiate is (3*t+2*e^t)*e^(-3t/2) . I have tried simplifying it into these functions (terms, expressions): [(3*t)*e^(-3t/2)+2*(e^t)(e^(-3t/2)) =3t(e^(-3t/2))+2(e^(-t/2))...
  46. M

    Initial value problem (Laplace)

    Homework Statement Use the Laplace transform method to solve the differential equation y" -4y' +4y = e^2t subject to initial conditions y(0)=1, y'(0)=0 Homework Equations The Attempt at a Solution s^2Y(s) -sy(0) - y'(0) - 4[ sY(s) - y(0)] + 4Y(s)= L (e^2t) Y(s) [s^2 -4s...
  47. I

    Initial value problem differentials, close to getting answer

    Homework Statement y" + 2y' + 10y=0 y(0)=1 y'(0)= -1 solve initial value problem Homework Equations e^( ~ + iu)t= e^ ~t (cos ut + i sin ut) The Attempt at a Solution i've gotten pretty far into the problem, but i just can't seem to get the correct final answer. I changed y" +...
  48. G

    Initial value problem using partial fractions

    (t+1) dx/dt = x^2 + 1 (t > -1), x(0) = pi/4 I have attempted to work this by placing like terms on either side and then integrating. 1/(x^2 + 1) dx = 1/(t + 1) dt arctan x = ln |t + 1| + C x = tan (ln |t + 1|) + C pi/4 = tan(ln |0 + 1|) + C pi/4 = C x = tan (ln |t + 1|)...
  49. M

    Solving an Initial Value Problem Using Euler's Method

    When solving the initial value problem y ' = 2y-x, y(11)=6 using Eulers method with h=0.2, y0=? I know how to solve Euler's equations with the formula yn = yn-1 + F(xn-1 + yn-1)(h), however I'm not quite sure how or what they want in this particular case. Can anybody please help me out if...
  50. B

    Undetermined coefficients + initial value problem

    Homework Statement Find solution for y'' + y' -2y = 2x with initial values of y(0) = 0, y'(0) = 1 Homework Equations I have found yc = c1*exp(-2x) + c2*exp(x), but finding yp is what I'm having trouble with... AND THEN I'm not so sure how to go about the initial value. The Attempt at...
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