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.
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##...
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:
$${5} \frac{\text{d}\theta}{\text{d}t}=-6\theta$$
$${5} \frac{\text{d}\theta}{\text{d}t} =\frac{\text{-6}\theta}{5}$$...
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...
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...
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...
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...
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 ?
(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.
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) =...
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...
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...
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...
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 =...
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...
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
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...
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...
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...
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...
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...
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?
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...
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...
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...
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...
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...
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...
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...
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...
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 =...
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...
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
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!
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...
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...
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?
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?
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...
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
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...
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
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...
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...
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...