Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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4
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1answer
39 views

Still getting wrong answer after trying to solve $x''(t)+4x(t)=t^2$ where $x(0)=1$ and $x'(0)=2$

I am trying to solve this differential equation: $$x''(t)+4x(t)=t^2,x(0)=1,x'(0)=2$$ The answer should be: $$x(t)=\frac{1}{4}t^2-\frac{1}{8}+\frac{9}{8}\cos{2t}+\sin{2t}$$ Which is also verified ...
0
votes
0answers
15 views

Why don't we check the exactness of differential equation with Inspection cases?

When solving the differential equations which are reducible to exact differential equations using Inspection cases for example: Solve: $2xy^2 + ye^xdx = e^xdy$ The integrating factor would $1/y^2$ ...
0
votes
0answers
7 views

Using the method of isoclines with logistic equation to create direction field

I am a little unsure on how to use the method of isoclines to model $\frac{dp}{dt} = 3p-2p^2$. As far as I know I need to set $3p-2p^2 = c$ where $c$ is the slope of the field on that line. When I set ...
1
vote
1answer
21 views

Lowering the order of a linear differential equation

Let $$L(x) \equiv x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}(t)x'+a_n(t)x=0.$$ and let the following solutions be given: $x_1,x_2,...,x_m(m<n)$- linear independent solutions. Let's find: $x_{m+1}, ...
0
votes
0answers
12 views

Terminology in DE, difference between Particular and Actual solution

Yesterday I started studying and preparing for a course in Differential Equations and today I came across something that confuses me; I watched a lecture on IVP and they used both Actual solution and ...
2
votes
1answer
32 views

The differential equation $\frac{dy}{dx} +y^2 + \frac{x}{1-x}y = \frac{1}{1-x}$

I am learning how to solve differential equations and making some progress. However, how can one solve this example? The task is to find the solution to the equation $$\frac{dy}{dx} +y^2 + ...
1
vote
4answers
68 views

The differential equation $\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;$

I am learning differential equations and can do the basic examples. However, how can you solve the differential equation $$\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;?$$
0
votes
1answer
36 views

Integrating Factor.

$(axy^2 + by) dx + (bx^2y + ax) dy=0$ I have asked this question before too, but i wish to know the method for evaluating the integrating factor which is $\frac {1} {(a-b)(x^2y^2-xy)}.$ So far i ...
1
vote
1answer
15 views

how to write a function in terms of Heaviside step function

I'm reading Paul Online Notes. There's an example of writing a function in terms of Heaviside step function as follows: $$ f(t) = \begin{cases} -4 &\text{if } t < 6, \\ 25 &\text{if } 6 \le ...
3
votes
0answers
22 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
0
votes
2answers
28 views

General ODE question

Find the general solution $y(t)$ of the ordinary differential equation $$y''+\omega^2 y=\cos \omega t,$$ where $\omega>0$. I'm relatively new to ODEs and PDEs but can someone show me the ...
0
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0answers
17 views

Love's equation $f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$

Let us consider Love's equation: $$f(x)+\frac{1}{\pi} \int_{-1}^{1} \frac{f(t)}{1+(x-t)^2}dt=1, \ \ (|x|\geq 1)$$ Is $f(x)$ a two times differentiable function?
0
votes
1answer
16 views

Find $\varphi_{1}$ from $q_{1}=A_{1}\sin(\omega t+\varphi_{1})$

I have $q_{1}=A_{1}\sin(\omega t+\varphi_{1})$ where $q_{1}=0$ and $\dot q_{1}=v_{0}$ and I must find $\varphi_{1}$. I know that $\varphi_{1}$ must be zero but I must demonstrate it first. ...
1
vote
1answer
29 views

How to solve differential equation $3p^2e^y-px+1=0$ ,$p =\frac{dy}{dx}$

How to solve differential equation $$3p^2e^y-px+1=0$$ where $$p =\frac{dy}{dx}$$ I have tried to solve for p and for x, but i am not getting anywhere. Can someone help me with this Thanks
0
votes
1answer
42 views

Method of successive approximations to solve y'=y^2

(a) Show that all the successive approximations for the problem $y'=y^2$, $y(0) = 1$, exist for all real $x$. (b) Find a solution of the initial value problem in (a). On what interval does it ...
2
votes
1answer
51 views

Is okay to have different solution to differential equation?

Suppose I have the following differential equation: $ydx - xdy - dx = 0$ Now, I could divide it by Integrating factor $x^2$ to get: $(xdy - ydx)/(x^2) - dx/x^2 = 0$ Use the inspection rule to get: ...
0
votes
0answers
19 views

Determine for what values of m the function is a solution

I was working through some differential equations and came across this problem. Determine for which values of $m$ the function $\phi(x)=e^{mx}$ is a solution to the given equation. A) ...
0
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0answers
10 views

Solve the following differential equation subject to the specified boundaries:

Solve the following differential equation subject to the specified boundaries: my answer please review my answer and correct it if it is wrong thanks
0
votes
1answer
19 views

Trigonometric Differential Equation 3

$(x\cos y-y\sin y)dy+(x\sin y+y\cos y)dx=0$ ATTEMPT: Rearranging the terms: $(x\cos ydy+y\cos ydx) -y\sin ydy+x\sin ydx=0$ Dividing by $\cos x$ we get: $(xdy+ydx)-y\tan ydy+x\tan ydx=0$ $ ...
3
votes
1answer
20 views

Solving general linear ODE $\sum_{k=0}^n y^{(k)}=0$

Is there a way to solve this general linear ODE: $$\sum_{k=0}^n y^{(k)}=0$$ For the first few $n$ here are the solutions: $$\begin{array}{c|c} n & y \\ \hline 0 & 0 \\ 1 & c_1 e^x \\ 2 ...
0
votes
1answer
26 views

Find the position function from the piecewise-defined velocity function

I am getting stuck on a position function problem in my Diff Eq class. Problem 22 is shown on the right in the picture below. On the left is the answer. My work below shows that I get stuck ...
0
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0answers
13 views

Study materials for Differential equations and Fourier analysis

In two days, on Monday, a new course called Introduction to differential equations starts, and when that ends in one month another called Fourier analysis and its application starts (Both are actually ...
4
votes
5answers
88 views

Solve: $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$

It is asked to solve the ODE $x''(t)-2x'(t) + x(t) = 2 \sin(3t)$ for $x(0)=10, \; x'(0)=0$ It is equivalent to the first order system in two variables $$\begin{bmatrix} x' \\ y' \end{bmatrix} = ...
4
votes
2answers
120 views

A singular Gronwall inequality

Let $f : [0,T] \to R^+$ be a continuous function such that $f(0)=0 $ and : $$ f(t)\le C\int_0^t s^{-1}f(s) ds,\; \forall t\in [0,T] $$ for some constant $C>0.$ Is it true that $f(t)=0,\; \forall ...
0
votes
1answer
32 views

Chain Rule in Polar coordinates

I was looking for an intuitive explanation for the total derivative in polar coordinates. Let me be somewhat more specific: Take a standard line of reasoning that the gradient w.r.t. polar coordinates ...
2
votes
5answers
93 views

Solve $y''=y^2$

Are there any 'basic' solutions to this differential equation (ie using polynomials, exponetials, trigonometric functions and logarithms)? I cannot figure it out at all using the techniques I know for ...
-1
votes
1answer
25 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
0
votes
1answer
41 views

Number of solutions of the differential equation ${dy}\over {dx}$=$y^{1/3}$ $y(0)=0$

The given differential equation is ${dy}\over {dx}$=$y^{1/3}$, $y(0)=0$ I got the solution $$y^{2/3}={{2}\over {3}}x$$ $$i.e. y^{2}={{8}\over {27}} x^{3}$$ $$i.e. y= \pm \sqrt{{{8}\over ...
2
votes
1answer
39 views

solution of 1st order PDE

Find the solution of PDE, $$u_xu_y = u$$ with the initial condition $u(x,0) = 0$ in the domain $x \geq 0$ and $y \geq 0$. I have try the method of characteristic, but it seems like not working for ...
2
votes
1answer
78 views

Exact Differential Equations

$M(x,y)dx + N(x,y)dy=0$ is said to be a perfect differential when $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$. Let $M_y=\frac{\partial (M(x,y))}{\partial y}$ and ...
1
vote
1answer
35 views

Sketching phase portrait of an ellipse

I have a system of linear ODE's as follows: $$\frac{dx}{dt} = y, \frac{dy}{dt} = -4x$$ which has solution $$\begin{bmatrix}x\\y\end{bmatrix} = \alpha\begin{bmatrix}\cos2t\\-2\sin2t\end{bmatrix} + ...
2
votes
1answer
34 views

Taylor Series General Formulas

I'm looking at 2 different Wikipedia pages: The formula here is different than the one given at the end of the section here. Aside from the remainder, why choose one over the other? I'm assuming ...
1
vote
2answers
38 views

Slope field of $y'=x^2 - y^2$

I don't know how I am supposed to go about creating a table with slope values for the graph so that I can sketch them. I knew how to do it when $y'$ equations had $y$ only or $x$ only, but not when ...
0
votes
1answer
19 views

Definition of `equivalent systems of linear differential equations'

I'm reading F.Beukers' `Notes on differential equations and hypergeometric functions', and I can't work out the details of something that seems obviously true. We have a field $K$ endowed with a ...
1
vote
1answer
40 views

Why is t used instead of delta t?

Consider a tank that holds $V$ liters of water. Let $x_0$ kg of salt be dissolved in the water at time $t_0$. Suppose that $V_o$ amount of the mixture is leaving the tank in every time interval, ...
1
vote
0answers
51 views

Why is $\frac{\partial }{\partial y}\int M dx = \int \frac{\partial M}{\partial y}dx$

$M$ is a function of $x$ and $y$. I'm getting this question from looking at the solution of the exact equation $M \mathrm{dx} + N\mathrm{dy} = 0$.
1
vote
1answer
31 views

Topics to master (be literate at) before differential equations?

Good evening, I'm really enthusiastic about learning differential equations because it was said that D.E. is the most important tool of mathematics "can be used for modelling real-world physical ...
0
votes
1answer
35 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
0
votes
0answers
30 views

Differentiating CDF

I'm trying to differentiate the cdf of z with respect to x where the upper bound is a function of x and z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} \Phi ^{\prime} (z) ...
0
votes
0answers
18 views

Finite difference for variable coefficient with Neumann Boundary

The equations is the same as this post, but with respect to the Neumann boundary. The physically correct boundary conditions for this equation are \begin{equation} A(x)\frac{\partial u(x)}{\partial ...
1
vote
2answers
41 views

Homogeneous 1st order ODE

This question comes from Schaums Calculus, CH59 Q18 which has had me confused for a couple of days now. Solve: $$ {dy \over dx} + y = xy^2 $$ I understand that this is a non-linear first order ode, ...
3
votes
0answers
15 views

Green's Functions: Solvable non homogeneous Sturm-Liouville with non homogeneous boundary conditions

I was just presented with this problem in my PDE Methods course which involves a non homogeneous Sturm-Liouville problem, which states as follows: Find the conditions under which the following SL ...
2
votes
1answer
24 views

Finding an equilibrium solution to a first order system of equations.

Given a model: $ y''+\alpha y'+\beta y + \gamma y = -g $ I can see that it can be converted to a system of first order equations as follows: $y_{1}=y$, $y_{2}=y'$ and as such $y_{1}'=y'$ and ...
1
vote
2answers
22 views

Difference between two solution of inhomogeneous linear equation

Show that the difference between two solutions of an inhomogeneous linear equation $Lu =g$ with the same $g$ is the solution of the homogenous equation $Lu=0$ I know the definition of linearity, but ...
2
votes
2answers
19 views

Help interpreting the solution for a differential equation

The differential equation is $\frac{dx}{dt} = x + x^2$ Solving for $x$, I got $x = (ce^t)/(1- ce^t)$ where, $c = x_0/(1+x_0)$ and $x_0$ is the initial value of $x$ at $t=0$ Now, the value of ...
0
votes
2answers
28 views

Differential Equations Pressure and Density derivation

The pressure $p$, and the density, $\rho$, of the atmosphere at a height $y$ above the earth's surface are related by $dp = -g \rho\; dy$. Assuming that $p$ and $\rho$ satify the adiabatic equation of ...
1
vote
1answer
31 views

Show that $m_1=\frac{m_2-\tan{(a)}}{1+m_2\tan{(a)}}$ [on hold]

Given: $m_1=\tan{(a_1)}$ and denotes the slope of the required family at some $(x,y)$ $m_2=\tan{(a_2)}$ and denotes the slope of the given family at the same $(x,y)$ it also gives the hint that ...
-2
votes
1answer
52 views

Solving Differential Equation $\frac{dy}{dx} = 1 -\sin(x+y)/(\sin y \cos x)$ by separating variables

Initial value is $y(\frac{\pi}{4})$. I got to $\frac{\mathrm{d}y}{\mathrm{d}x} = 1 - \frac{\sin(x) \cos(y) + \sin(y) \cos(x)}{\sin(y)\cos(x)}$ by using the $\sin(x+y) = \sin(x) \cos(y) + \sin(y) ...
1
vote
0answers
36 views

Diagonalization: Differential Equations

The booking being used for this course is Differential Equations and Dynamical Systems by Lawrence Perko. The problem is as follows: Let the $n\times n$ matrix $A$ have real, distinct ...
1
vote
1answer
24 views

Uncoupled Linear System: Differential Equations

I'm trying to make sense of a problem I was given in class and I want to know if I am on the right track. The question is as follows: If $\vec{u}(t)$ and $\vec{v}(t)$ are solutions of the linear ...