Differential equations Definition and 999 Threads

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy.

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  1. P

    2nd order nonhomogeneos differential equations with initial conditions

    Homework Statement The problem states d^2y/dt^2 +15y= cost4t + 2sin t initial conditions y(0)=y'(0)=0 Homework Equations The Attempt at a Solution All I have is this r^2+15=0 making r(+-)=√15 and making yh= C1cos√15+C2√15 the next part includes solve for...
  2. P

    2nd order nonhomogeneos differential equations with initial conditions

    I have a problem which in involves a second order differential equations with imaginary roots and I can seem to know how to finish the problem. d^2y/dt^2 +15y =cos 4t+2 sin t this is what I got so far r^2+15=0 for the homogeneous part r=+-(√15) Yh=C1cos√15+C2sin√15 now is...
  3. G

    Differential Equations behavior for large t?

    dy/dt = 2 - 2ty y(0) = 1 I am not asked to solve this (I know it's not easy to solve), but what I am asked is, "for large values of t is the solution y(t) greater than, less than, or equal to 1/t"? I would think less than because 1/e^(t^2) converges faster than 1/t, but at the same...
  4. Q

    Solving Non-Homogeneous Differential Equations with Two y' Terms

    Homework Statement I have to find the differential of (y-xy')^2=x^2+y^2.Now,I have solved hom. equations but this is different because there are two y'. I know how to prove that it is a hom. equation of degree zero, so we can skip that, but how to solve this? Some hints would be highly...
  5. F

    Deriving differential equations for free rotation

    I was asked to formulate the equations governing the rotation of a body moving without any external moments acting about its centre of mass in terms of a coupled system of first order, nonlinear differential equations. I decided to go with the Euler equations, and I ended up with this...
  6. T

    Repeated Eigenvalue of a n=3 system of differential equations

    Homework Statement x' = \begin{pmatrix}0&1&3\\2&-1&2\\-1&0&-2\end{pmatrix}*x The Attempt at a Solution I've found the repeated eigenvalues to be λ_{1,2,3}=-1 I've also found the first (and only non zero eigenvector) to be \begin{pmatrix}1&2&-1\end{pmatrix}, but I'm not entirely...
  7. M

    Solving Partial Differential Equations with Laplace Transform

    Homework Statement \dfrac{\partial^2 \varphi }{ \partial x^2} - \dfrac{\partial ^2 \varphi }{\partial t^2} = 1 Initial Conditions: \varphi (x, 0) = 1; \varphi_t (x, 0) = 1 Boundary Condition: \varphi (0, t) = 1 On 0 \leq x < \infty, 0 \leq t < \infty...
  8. N

    Complex differential equations to find functions [TRIED]

    Homework Statement I need to solve two differential equations to find a population function P(t). I am able to do this with problems like Newtons law of cooling: dT/dt=-k(T-Ta) solves to: dT=-k(T-Ta) dt ∫1/(T-Ta)dT=∫-k dt Ln(T-Ta)=-kt e-kt=T-Ta T=Ta+E-kt However I have been...
  9. M

    Solving Rocket Launch Differential Equations

    Hi I'm currently working on a project which involves solving the rocket launch differential equations to find the apogee of an orbit. I know the analytical model for the equations as: Δu = Isp*g0*ln(mf/me), where Isp is fuel impusle, mf is mass of full tank and me is mass of empty tank, but...
  10. G

    Differential Equations - Second Order

    Homework Statement Hi, It's been a while since I have taken differential equations. How do I solve an equation like this: k_{1}\frac{d^{2}V_{x}(t)}{dt^{2}}+k_{2}\frac{dV_{x}(t)}{dt}+k_{3}V_{x}(t)=0 Homework Equations The Attempt at a Solution I have looked through my...
  11. N

    How Does the Integrating Factor Simplify Differential Equations?

    im in calculus 2 right now and we are doing differential equaitons right now. I am confused as to why when i find the integrating factor I(x)=e^(∫p(x) and when i multiply both sides i get e^∫(p(x))[(dy/dx)+p(x)*y]=d(e^(p(x)dx)*y) how are they equal. i will give an example. (dy/dx)+y=x*e^(x)...
  12. B

    Vibrations and differential equations.

    Homework Statement Suppose the motion of a spring has natural frequency 1/2 and is undamped. If the weight attached is 32lb, write a differential equation describing the motion. Homework Equations my''+ky=F_ocosωt 32y"+8y=? ω_o= (k/m)^.5 The Attempt at a Solution → .5=(k/32)^.5 → k=8...
  13. W

    General question regarding solutions to differential equations.

    My textbook defines a solution to a differential equation to be a function f(x) such that when substituted into the equation gives a true statement. What I'm confused about are singular solutions. For example the logistic equation: dP/dt = rP(1-P/K) where r and K are constants. My textbook says...
  14. G

    Solving differential equations (circular motion)

    Homework Statement I have a differential equation of the form \frac{dZ}{d\theta} + cZ = a cos \theta + b sin \theta Where Z = \frac{1}{2}\dot{\theta}^{2} I need to find the general solution of this equation. a, b and c are all constants. Homework Equations The questions suggests using...
  15. N

    Solutions to differential equations

    y(x)=A*e^(λx) ; y'=λy attempt at solution: y'(x)= Ae^(λx)*λ λy= Ae^(λx)*λ divide by λ, which cancel. then i get: y=Ae^(λx) i want to say the differential equation holds but the issue i see is that y' and y'(x) are not equal derivatives, so my final answer is that the...
  16. J

    Differential Equations: Stuck on one step of H.L.D.E problem

    I need to find a,b,c,d, and e. I know how to do these problems the normal way but now he's giving us the answer and asking us to work backwards. I'm stumped. I think I night need to use some sord of system of equations but I'm not sure what it would look like..
  17. A

    A system of 200 second order differential equations

    How to solve a system of 200 second order differential equations in matlab? I know simulink can work but it takes damn long time to draw those blocks for 200 equations...
  18. T

    Dealing with Random Coefficients in Linear Differential Equations

    How does one go about dealing with a linear differential equation with random but constant coefficients (e.g. X''(t) + A*X'(t) + B*X(t) = 0 where A and B are random variables, but are constant with time)? I've searched for things like random differential equations and stochastic differential...
  19. A

    Mean Value Property in partial differential equations

    Homework Statement S is a ball of radius 1 in R^2; Δu=0 in S u=g in ∂S, g(x1,x2)>1 for any (x1,x2) in ∂S. Show that for any r satisfying 0<r<1 there is a point (x1,x2) in S such that u(x1, x2) >=1. Homework Equations using mean value formula: ∫u(y)dy=1/Vr^n(∫u(y)dy) The...
  20. R

    Linear system of differential equations with repeated eigenvalues

    Homework Statement X'=AXA=\left[\begin{matrix} 0 & 1 & 0 \\ -1 & 0 &0 \\0 & 0 & -1\end{matrix}\right] Homework Equations n/a The Attempt at a Solution The eigenvalues are -1, and \pm i. I also can see that the matrix A is already in the form A=\left[\begin{matrix} \alpha & \beta & 0 \\...
  21. D

    Differential Equations Problem

    Homework Statement This is actually a question we went over in class, but I kind of spaced out when my teacher was explaining it. I have since solved it myself while I was reviewing for my calculus test, but when I compared my answer with the answer key my teacher provided to our class, I...
  22. M

    Object Dropped From 500m: Solved with Differential Equations

    an object is dropped from a height of 500m. when will object reach the ground level and with what speed? important: the solution must be by: Differential Equations.
  23. N

    Differential Equations: Salt Concentration and logs

    Homework Statement A tank initially contains 180 gallons of water in which 8grams of salt are dissolved. Water containing 9 grams of salt per gallon enters the tank at the rate of 3 gallons per minute, and the well mixed solution leaves the tank at the rate of 1 gallon per minute. The...
  24. A

    Solving a System of Differential Equations: Ants at the Corners of a Square

    Suppose there is an ant at each of the four corners of a square with side length 1, such that (0,0) and (1,1) are at opposite corners of the square. The ant at (0,0) is facing the ant at (1,0) is facing the ant at (1,1) and so forth. Each ant will choose its path such that it is always facing...
  25. 1

    Differential Equations and a vector analogy (weird question)

    Suppose I have a third order differential equation, and have three solutions, y1, y2, y3. I can check to see if they are linearly independent as such: if their wronskian is non-zero, they are linearly independent. But the wronskian is just a determinant of a matrix. y1 y2 y3 y1' y2' y3' y1''...
  26. rakeru

    Should I Take Calculus 3 and Differential Equations at Once?

    Hi! I'm a student at a community college and I'm getting an AA in physics to transfer to a university. I eventually want to apply to medical school with a bachelors degree in physics. I've recently been going over the classes required for medical school and for the physics major. I've...
  27. J

    Differential equations question

    Homework Statement I need to solve for a general solution of the following equation: (x^2+6x+12)dy=y^2dx 2. The attempt at a solution I tried using the method for seperable ODEs and got the following: ∫(x^2+6x+12)dy=∫(y^2)dx (x^2+6x+12)y=(y^2)x+C (x^2+6x+12-C)/x=y
  28. B

    Manipulation of Arbitrary Constants in Differential Equations

    Homework Statement yy''+(y')2 = 0 Homework Equations yv(dv/dy)+v2=0 The Attempt at a Solution Variable separable when solving for the first step the result is: - ln |v| = ln |y| + C1 Now, I've looked at the remainder of the solution with a few other sources and the cause of...
  29. H

    Solving differential equations

    Homework Statement solve (y^15)x(dy/dx)=1+x using the initial condition y(1)=3 express y^16 in terms of x Homework Equations The Attempt at a Solution i change the equation to the form y^15 dy = (1+x)/x dx integrating both sides i got y^16/16=x+lnx+C To solve...
  30. N

    (Differential Equations) General and singular solutions

    Homework Statement Find the general solution and any singular solutions to (2xy^3+4x)y'=x^2y^2+y^2.Homework Equations The Attempt at a Solution 2x(y^3+2)y'=y^2(x^2+1) \int\frac{y^3+2}{y^2}\,dy=\int\frac{x^2+1}{2x}\,dx \frac{y^3-4}{2y}=\frac{x^2+2\ln x}{4}+C Is this correct? To find the...
  31. J

    Differential Equations Mixture problem

    at t=o an evenly mixed 40 liters solution with 100grams of salt is poured into a container. solution comes in at 2 grams per liter at 3 liters per minute and flows out at the same rate. At what time is the salt solution 120 grams? I got -9.something minutes for my answer :/ This question was...
  32. S

    Differential Equations dealing with spring physics

    I am greatly struggling with a homework assignment given out by my physics professor. It mostly differential equations but based on spring physics. I'll type out the first couple parts but will most likely need help with more as I get farther. Homework Statement Y(t) : the y position of mass...
  33. T

    Programs Differential Equations or Matrix Algebra for Physic Major

    I am signing up for my third quarter of college classes soon and I have to choose if I am going to take Differential Equations or Matrix Algebra this quarter. I am given the option to take either of them and I do not know if I will be able to take the other one anytime soon. Which of the two...
  34. A

    Differential Equations Euler's method

    Find the solution y = φ(t) of the given problem and evaluate φ(t) at t = 0.1, 0.2, 0.3, and 0.4. 1.y'=3+t-y y = φ(t)=t-2e^-t y(1)= 0+(0-2e^0)*(.1)=.8 and the correct answer is 1.19516 2. y'=2y-1 What I'm getting stuck on is do I use the formula y(n)=y(n-1)+f(t(n-1),y(n-1)h because...
  35. F

    Differential equations by series and also by an elementary method

    Homework Statement Solve the following differential equations by series and also by an elementary method and verify that your solutions agree. (x^2+2x)y''-2(x+1)y'+2y=0 Homework Equations y=\sum_{n=0}^{\infty} a_nx^n y'=\sum_{n=1}^{\infty} na_nx^{n-1} y''=\sum_{n=1}^{\infty}...
  36. B

    Differential Equations Problem, logistic models

    Homework Statement Given that a population, P, after t months, can be modeled by the logistic model dP/dt = .3 P (3.5 - P/40). P(0) = 30 a) Solve the diff eq b) Find the population after 2.5 months c) Find lim P(t) as t -> infinity Homework Equations P(t) = P0 P1 /(P0 + (P1...
  37. P

    Radiation and differential equations

    Sam is seeping into a room from the basement at a rate of $$\frac{1}{6} \times 10^6 \frac{\mathrm{pCi}}{\mathrm{hr}}$$ (pCi=picocuries). The room contains $$10^6$$ liters of air. (The rate was chosen so that the room reaches the EPA action level of $$4 \frac{pCi}{liter}$$ after $$24$$...
  38. X

    Differential Equations in Matrices

    I realize that Δ(s) is the cross product of the matrix on the left, but how did the solutions manual get the matrix on the far right multiplied by R_1(s) and R_2(s)? I need those matrix values to do the rest of the problem. Any help is appreciated, thank you.
  39. L

    Differential equations - cannot solve one

    Greetings everyone. Can you help me please with solving this differential equation? \large \frac{dv}{dt} = a_g + \alpha v^n where \Large a_g \alpha n are constants. Artelnatively with specific n as n = 1, 2. I have no idea what to do... Thank you very much
  40. N

    Differential Equations and Initial Values

    Homework Statement If y is the solution of the initial value problem y'(t) + y(t)/t = 1/t4 y(2)=1 What is y(3) Homework Equations The Attempt at a Solution I u(t) = e∫1/t dt = eln t = t ∫[yt]' dt = t-4 dt yt = -1/3t3 + C Plugging in my initial value I solve for...
  41. marellasunny

    Solving differential equations using surface intersections

    Take for example a system of differential equations: dx/dt=f(x,y) dy/dt=g(x,y) If I plot the surfaces for f(x,y) and g(x,y) and find their intersections,what does this symbolize in terms of the differential equations? I'm thinking equilibrium points(??) Can one go further and find the...
  42. K

    First Order Differential Equations where a<x<b (Intial value?)

    Hi, I'm having trouble understanding what to do when a First order equation has an inequality at the end of it. For example : sqrt(y-x^2y)*dy/dx = -xy where -1<x<1 I've solved the differential equation with y = 1/4(2C*sqrt(1-x^2) + C^2 -x^2 +1) where C is a constant. What do I do with...
  43. X

    Frobenius method for a differential equations

    Homework Statement The function satisfies the differential equation f''(x) = xf(x) and has boundary conditions f(0) = 1 and f'(0) = 1 Use Frobenius method to solve for f(x) with a taylor expansion of f(x) up to the quartic term a4x4 Homework Equations f(x) = a0 + a1x + a2x2 + a3x3 + a4x4...
  44. N

    Differential Equations: Largest rectangle on ty-plane

    Homework Statement Find the largest open rectangle in the ty-plane that contains the initial value point and satisfies the following theorem: Let R be the open rectangle defined by a<t<b, α<y<β. Let f(t,y) be a function of two variables defined on R where f(t,y) and the partial derivative...
  45. N

    Differential Equations and Radioactive Decay

    Homework Statement The radioactive decay of a substance is proportional to the present amount of substance at any time t. If there was 15 grams at t=0 hours and 10 grams at t=3 hours. Set up the differential equation that models this decay and use the method of separation of variables to solve...
  46. N

    Solving Differential Equations

    Homework Statement A psychology class is studying memory. Several objects are uncovered to view for a given amount of minutes and then covered again. At most 150 objects can be viewed and remembered. The class found that after 10 minutes the average student could remember 25 objects. The...
  47. 3

    Two coupled, second order differential equations

    While studying the derivation of the normal modes of oscillation of a liquid sphere in the paper "Nonradial oscillations of stars" by Pekeris (1938), which can be found here, on page 193 and 194 two coupled second order differential equations in two variables are merged into one fourth order...
  48. N

    Differential Equations: If y(1) =3 then y(1/2) = ?

    Homework Statement If y satisfies the differential equation: ty'(t) + tln(t)y(t) = 0 and y(1) = 3 then y(1/2) = ? I have attached an image of the question with the possible solutions: Homework Equations The Attempt at a Solution Initially I tried plugging in t=1 and...
  49. M

    Mathematics - Q 8 - Forming Differential Equations

    mathematics - Q 8 -- Forming Differential Equations
  50. C

    Differential Equations without numbers

    Homework Statement Here is the whole problem. Homework Equations Not sure The Attempt at a Solution As you can see I got the first part. I solved the equation at part a.) for A(c) and plugged it into the D.E. and then solved for B(c). I need help on how to do the next part. How...
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