What is First order differential: Definition and 103 Discussions

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. Solve the given first order differential equation

My thinking is two-fold, firstly, i noted that we can use separation of variables; i.e ##\dfrac{dy}{y}= \sec^2 x dx## on integrating both sides we have; ##\ln y = \tan x + k## ##y=e^{\tan x+k} ## now i got stuck here as we cannot apply the initial condition ##y(\dfrac {π}{4})=-1##...
2. Engineering First order differential equations (movement of a rotary solenoid)

My question i am trying to solve: I have successfully done first order equations before but this one has got me a little stuck. My attempt at the general solution below: $${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$ $${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
3. Solve the first order differential equation

From my working...I am getting, ##xy=####\int x^{-1/2}\ dx## ##y##=##\dfrac {2}{x}##+##\dfrac {k}{x}## ##y##=##\dfrac {2}{x}##+##\dfrac {6}{x}## ##y##=##\dfrac {8}{x}## i hope am getting it right...
4. How Do You Solve the Differential Equation dy/dx = 1 - y^2?

This is the question; This is the solution; Find my approach here, ##x####\frac {dy}{dx}##=##1-y^2## →##\frac {dx}{x}##=##\frac {dy}{1-y^2}## I let ##u=1-y^2## → ##du=-2ydy##, therefore; ##\int ####\frac {dx}{x}##=##\int ####\frac {du}{-2yu}##, we know that ##y##=##\sqrt {1-u}## ##\int...
5. MHB First order differential equations

Hi, Is the answer: y(x) _homogenous =v(x) y(x) _private =u(x)v(x) ? Or they refer to something else? I don't know how to approach to it
6. Solving a first order differential equation with initial conditions

Hello! Consider this ODE; $$x' = sin(t) (x+2)$$ with initial conditions x(0) = 1; Now I've solved it and according to wolfram alpha it is correct (I got the homogenous and the particular solution) $$x = c * e^{-cos(t)} -2$$ and now I wanted to plug in the initial conditions and this is...
7. First order differential equation involving a square root

Summary:: solution of first order derivatives we had in the class a first order derivative equation: ##\frac{dR(t)}{dt}=-\sqrt{\frac{2GM(R)}{R}}## in which R dependent of time. and I don't understand why the solution to this equation is...
8. I Where does the exponential function come from in roots?

For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?
9. I Solving a nonlinear first order differential equation

(a'[t]/a[t])^2 == K*(A + B*a[t]^-6)^1/2} is the equation to be solved for getting the solution of a(t) in terms of time(t). Any ideas on how to solve this problem? Use of Matlab or Mathematica is accepted.
10. First order differential equation

Homework Statement Solve the following differential equation such that ##x(0)=1##. ## \dfrac{dx}{dt} + 2tx = 3e^{-t^2}+t## Homework Equations Integrating factor: ##\mu(t) = exp\left(\int_0^t2t \right)## The Attempt at a Solution I used the integrating factor and then got the solution ##x(t) =...
11. Wondering if these two First Linear Order IVPs are correct

Homework Statement I am having trouble proving if the equation i have found for number 1 is correct. I have posted my solution to get back to the main problem in the first photo below. For number 2 I am having trouble isolating for 1 y(x). Did i do the integration and setup properly?Homework...
12. B

Choosing Axis of Rotation in Cylinder Oscillation Problem

Homework Statement Here is a problem we worked in class. I already know the answer, just had a question on the method. Two cylinders are connected with by a small rod (with presumably negligent mass) through their centers. The cylinders can roll freely. A spring is attached to the small rod...
13. Convert second order of diff. equations to first order

Homework Statement I have this set of equation: My''+Cy'+Ky=0 but C=0 M is a matrix consist of {(-m) (0)/( -1/12mb^2) (-1/12mb^3)} and K is a matrix of {(-K1-K2) (-K2b)/ ((K1b-K2b)/(2)) (-K2b^2/2)} and y is a coordinate system which is (x1,θ) Now i have to convert these...
14. Calculus First order differential equations ?

what is a good book to learn first order differential equations ??
15. I First order differential equations ?

what is a good book to learn first order differential equations ??
16. A non-exact nonlinear first ODE to solve

Homework Statement Solve the following equation. Homework Equations ( 3x2y4 + 2xy ) dx + ( 2x3y3 - x2 ) dy = 0 The Attempt at a Solution M = ( 3xy4 + 2xy ) N = ( 2x3y3 - x2 ) ∂M/∂y = 12x2y3 + 2x ∂N/∂x = 6x2y3 - 2x Then this equation looks like that the integrating factor is (xM-yN). IF =...
17. A Nonlinear first order Differential equation

I need to solve the well known momentum equation in 3D cylindrical coordinates: ρ(∂v/∂t +(v.∇)v)=A where A and the velocity v are both local vector variables. I am actually looking for the stationary solution to the equation, i.e. no ∂/∂t term) I have tried evolving the velocity and tried...
18. First order differential equation

Homework Statement dy/dx = (x +y) / (x-y) , i am asked to find the first order differential equation , but the ans i gt is different from the ans given Homework EquationsThe Attempt at a Solution
19. Solving a first order differential equation

Homework Statement Solve the differential equation: dy/dx = 2/(x+e^y) Homework EquationsThe Attempt at a Solution I tried to use the substitution v=x+e^y, but I didn't get very far: v’=1+e^y y’ v’-1=(v-x)y' y’ = (v’-1)/(v-x) (v’-1)/(v-x) (x+v-x)=2 V (v’-1)/(v-x)=2 vv’-v=2(v-x) vv’-3v=-2x...
20. I First Order Differential Equation

Basically, I am confused by one question in a practice paper in which the equation is given as follows: dy/dx = e^-2y and I know the general solution is equal to : y = -0.5e^-2y + C which would make sense if it was direct integration however it seems to me it is in fact separable...
21. Solving a first order differential equation

Homework Statement We have the equation ## (\frac{dr}{ds})^2+(\frac{l}{r})^2=1 ## and want to solve to get ## r=\sqrt{l^2+(s-s_0)^2}## Homework EquationsThe Attempt at a Solution I have worked backwards, plugging in the solution to prove that it is correct, but the closest I have gotten to...
22. Variation of parameters (1st order)

Homework Statement Find the general solution of the following equation: u(t): u' = u/t + 2t Homework Equations y' + p(x)y = Q(x)....(1) yeI = ∫ dx eIQ(x) + constant.....(2) The Attempt at a Solution I rearranged the equation to give: u' - u/t = 2t Then I considered the following...
23. First order differential equations and constants

Homework Statement ##y' = \frac{cos x}{sin y}## ##y' = \frac{6x^2}{y(1 + x^3)}## Homework EquationsThe Attempt at a Solution So I was working through some textbook problems and there's something about the solutions of the above equations I don't quite understand. The first one: ##\int sin...
24. Why Does a Constant Appear Only on One Side in This Differential Equation?

I don't understand this first order differential equation: https://lh5.googleusercontent.com/UUpQF4YjmjJRPvFuzGg2MhpMMMDyi2KFZPCKMKVIXGREc1owvXDzGR0bcA=s600 How is it possible to get an exponent as answer?
25. Help with First Order Differential

Homework Statement The Attempt at a Solution The first part is fairly simple I think. It's just rate of accumulation = rate of generation - rate of output(losses) I'm not too sure how to solve this differential equation. I divide the whole equation through by mc and rearrange but I keep...
26. Linear first order differential equation with unknown function

I was wondering if there is a way to get specific numerical values for the following differential equation: f'(x)+ \frac{1}{x-20}\cdot f(x)=\frac{1}{x-20}\cdot g(x) I have numerical values for g(x) for about 10 different x values. I need to find f(x) numerically for those same values...
27. Yet another first order differential equation

Homework Statement okey, so i got stuck at another step in the way of solving de's.I've been studying DE of this form: y' + P(x)y = Q(x) Homework Equations The Attempt at a Solution So, first we solve y' + P(x)y=0 for y. \frac{dy}{y} = -P(x)dx , we integrate this and get...
28. First order differential equation

Homework Statement I'm starting college this autumn(physics) and I started learning some calculus on my own, basic stuff like first order differential equation and so on.Recently i stumbled on something that i don t understand.I was reading the course and re-solving the given examples for...
29. Solving First Order Differential Equation

Homework Statement I have been trying to solve this equation but keep coming to the same solution, which according to my book is not the correct one. Is anyone able to point out what I am doing wrong? \frac{dy}{dt}-\frac{1}{2}y=2cos(t) The Attempt at a Solution To solve, use...
30. First order differential eqn dy/dx + Py = Qy^n

show that the substitution z = y^-(n-1) transforms the general equation dy/dx + Py = Qy^n, where P and Q are functions of x, into the linear equation dz/dx - P(n-1)z = -Q(n-1). (Bernoulli's equation) Well, I looked up Bernoulli's stuff on internet, found the usual air flow equation but not...
31. First order differential equations

Homework Statement Consider the first order differential equation \frac{dx(t)}{dt} + ax(t) = f(t), x(0) = x_{0}, t\geq0 Suppose the "input signal" f(t)=e^{-t}, t\geq0 . (a) Find the solution to the equation. Find a condition on the parameter a so that the solution of the (forced) system...
32. MHB Non-linear first order differential equation question

Kindly solve it and i need help to understand NON LINEAR Questions
33. MHB First Order Differential Equations, given initial value....

I'm having trouble with this problem... I am almost certain that I have the first part correct which is solving the first order DiffEQ using an integrating factor. I think that I am computing the constant incorrectly. I have followed all steps, including the similar problem given on WileyPlus...
34. First Order Differential Equation

Homework Statement Find the general solution to the following differential equation. dy/dx = 2x( (y^2) + 1) Homework Equations The Attempt at a Solution I got all x terms on one side and all y terms on the otherside 2x dx = 1/( (y^2) + 1`)dy integrate x^2 + c =...
35. MHB Verifying solution to first order differential equation

Verify the indicated function y=phi(x) is an explicit solution of the given equation. Consider the phi function as a solution of the differential equation and give at lease one interval I of definition. (y-x)y'=y-x+8 where y=x+4\sqrt{x+2} So the derivative is y'=1+\frac{2}{\sqrt{x+2}} and the...
36. MHB Help rearranging a linear first order differential equation

Hi I am trying to solve dy/dx = 3x^2-2x+2+(8/x *y) Can anyone show me how to rearrange to standard form as I am mightly confused :(
37. MHB Solving a first order differential equation

cosx(dy/dx) + ysinx = sinx cosx (dy/dx) + y/cosx = 1 e^integral (1/cosx) ? I feel like this has to do with ln again but not sure
38. MHB Solving a linear first order differential equation

4y'=e^(x/4) + y First I need to divide through by 4 correct? To obtain y'=(e^(x/4))/4 + (y/4) But then when I try to find integrating factor I just come up with e^(x/4) which I think is incorrect
39. First Order Differential Equation Help

Homework Statement dy/dx = 3 - 6x + y - 2xy Homework Equations dy/dx + p(x)y = c p(y) dy = q(x) dx The Attempt at a Solution Just realized where my mistake was, sorry!
40. 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...
41. Solving a First Order Differential Equation with Initial Conditions.

Homework Statement Solve the initial value problem: t(dy/dt)+8y=t^3 where t>0 and y(1)=0 Homework Equations None? The Attempt at a Solution It's a linear equation, so rearranged to dy/dt+8y/t=t^2. Took the integrating factor e^(∫8/tdt)=t^8 and multiplied through...
42. First order differential equation question

The problem is : dy/dx=(x(x^2+1))/4y^3 when y(0)=-1/√2 This is my work so far: ∫4y^3dy=∫x(x^2+1)dx (y^4)/2=((x^2+1)^2)/2+c The answer from the textbook is y=-(√(x^2+2)/2) As you can see, my work will never equal the textbook answer when you put it in the y= stuff form. What did I do wrong?
43. First Order Differential Equation with Reflected Argument

I am trying to solve: (x + 1 + f(-x) )(1 - f ' (x) ) = x+1 f(0) = x_0 x in (-1,1) I approximated it numerically but any analytic method I try fails. Any ideas?
44. Solving a first order differential equation

hi the differential equation i am attempting to solve is: \frac {dP} {dx} = \frac {gP} {1+P/Psat} here is what I have done: \frac {dP} {dx} = \frac {gP*Psat} {Psat+P} divide both sides by \frac {Psat+P} {gP*Psat} to get: \frac {Psat+P} {P*Psat} \frac {dP} {dx} =g...
45. Help Thrid second and first order differential equation

Help! Thrid second and first order differential equation! I have no idea how to accomplish this problem. If anyone knows help please help me solve this example before I take my test! Solve y''' - y'' - y' + y - x = 0
46. Can’t get a start on solving a first order differential equation

Homework Statement Am looking at a first order, non linear differential equation. I want to solve for a general solution and will have to find a constant with given parameters. Equation: 3x2y+8xy2+(x3+8x2y+12y2)dy/dx=0 Parameters: y(2)=1Homework Equations ∂M/∂y=∂N/∂x (Exactness) & (My-Nx)/N...
47. Solution of two first order differential equation with one algebraic equation.

Dear Friends i am trying to solve two first order differential eqs. with one algebraic eq. i am able to get solution of problem by simply solving two first order differential eqs. i do not know how to incorporate algebraic eq with my solution. please see attachment thanks in advance Ricky
48. Categorize first order differential equation

Homework Statement I'm trying to determine which categories various first order differential equations fall into (and once they're categorized they're nice and easy to solve). My list of categories is the following; linear equations, homogenous equations, bernoulli equations, exact equations...
49. First order differential equations.

Hi, I have problems rewriting equations with the term y'' as a system of first order differential equations. I've been given several equations and was told to write them as 1st order DEs, then calculate the numerical solution using the Euler's modified method. I know that y'=f(x,y), so if...
50. Coupled first order differential equation.

Homework Statement \frac{dx}{dt}=\gamma y \frac{dy}{dt}=-\gamma x solve for x and y Homework Equations The Attempt at a Solution I know how to solve it by substitution(without using matrix) I know how to solve a coupled second order differential equations in matrix form, but not...