1
vote
3answers
66 views

Solve y''' = y with given conditions?

I'm given the differential equation: $$y''' = y$$ which solves to: $$y(x) = c_1e^x + e^{-x/2}\left(c_2\cos\left(\frac{\sqrt{3}x}{2}\right) + c_3\sin\left(\frac{\sqrt{3}x}{2}\right)\right)$$ But I'm ...
2
votes
6answers
163 views

How to solve $y''' = y$

I'm trying to solve the following differential equation $ y''' = y$ and given conditions: $ y(1) = 3$, $y'(1) = 2$ and $y''(1) = 1 $ I began by making it: ...
1
vote
2answers
79 views

Solving ODE containing matrices

We have an ODE $ \psi'(t)_{_{3 \times 3}}=\psi(t)_{3 \times 3}(A_{3 \times 3}+B_{3 \times 3}t)\tag 1$ Given Data in Question We have no quarentee that $\psi'(t),\psi(t)$ both have inverse A,B are ...
0
votes
0answers
29 views

How do I solve this calculs problem [on hold]

a) Find the general solution of $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 0.$$ b) Solve $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 8\cos 2t + 6\sin 2t.$$ with $y(0) = 4$, $y'(0) = 0 $ How ...
0
votes
0answers
21 views

Differential equation Worded Problem [duplicate]

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
0
votes
1answer
17 views

Differential equation with modulus

I have a problem with $$y-xy'=(ln|x|+1)y^2 $$ because I do not know how to deal with the absolute value. I divide $\frac{y - xy'}{y^2}=ln|x| +1$, then substitute $t' = (\frac{x}{y})'$ get ...
1
vote
1answer
22 views

Differential equation of inclined plane

I'm having some trouble with the equation $$\frac{d}{dt}\dot{x}=g\sin\Theta \implies \dot{x}(t)=\dot{x}(t=0)+\int_0^t dt'\:g\sin\Theta=\dot{x_0}+g\:t\sin\Theta $$ which appears in page 4 of ...
0
votes
2answers
40 views

Find the functions $f$ that satisfy the given initial value problems

(a) $f'(x)+3x-2=0$, $f(2)=0$ (b) $2f'(x)-\sqrt{x^3} = 0$, $f(0) = 3$ I know the functions need to be integrated to find $f(x)$, however I am unsure as to how to integrate $f'(x)$ in the ...
0
votes
1answer
25 views

Calculus 7th Ed (Stewart) - Chapter 4 solution 2 page 332

This can be really ridiculous for you but I can't understand why dx is up on the root in solution 2 Shouldn't be "du = root(2x+1)*dx" instead of what is show below? Best Regards,
0
votes
1answer
39 views

Solving $\int \frac{1}{x-1}dx$ in two ways.

I have some confusion with this integral $$\int \frac{1}{x-1}dx$$ I can see the solution is $ln(x-1)$ However if I multiply the top and bottom by $-1$ I get $$\int \frac{-1}{1-x}dx$$ And then ...
-1
votes
0answers
23 views

Converting partial DE to integral Equation [closed]

Can anybody help me solving the below problem: What would be the functional corresponding to the following problem: $$ \frac{\partial ^{2}u}{\partial x^{2}}+ \frac{\partial ^{2}u}{\partial y^{2}} = ...
1
vote
2answers
66 views

Explain how to get the right solution of y $dy/dx=y$

When solving the following equation to find y as a function of x: \begin{equation} dy/dx=y \end{equation} First I divide both sides by $y$ and multiply both sides by $dx$: $dy/y=dx$ Then I ...
0
votes
0answers
38 views

how we integrate function s respect to y?

I would like to integral this function $$ \int\exp\big(x^2-y^2\big) \Big( \!2y\cos(2xy) +2x\sin(2xy)\Big)\mathrm{d}y $$ Thank you! $\exp\big(x^2-y^2\big)$ is a common factor for sin and cos
1
vote
0answers
39 views

Differentiation of an integral with respect to time variable

I have a little doubt about an integral that I try to differentiate according to time $t$. I have tried to do it with Leibniz rule but it did not work. Here it is ; ...
1
vote
1answer
55 views

Autonomous differential equation

Let $f: \Bbb R \to \Bbb R$ and $x_0 \in \Bbb R$, such that $f(x_0)> 0 $, and assume that $x(t)$ is the solution of $x'=f(x)$, such that $x(0)=x_0$. If $f(x) > 0$ then $x(t)$ is defined for all ...
0
votes
3answers
40 views

Solving using integrating factor [closed]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
0
votes
0answers
13 views

Integral formulation for LDE

I am trying to put the system in a integral formulation. All goes well for the first integration as I obtain What I don't know is how to perform the second integration in this last term. My ...
3
votes
1answer
51 views

Integral of [(1+2y^2)/(3-y)]dy (obtained from a differential equation)

This question actually arises from this Differential Equations question: Find the family of solutions for: $\displaystyle(1+2y^2)\frac{dy}{dx} + (3-y)\cos x = 0$ I ruled out the methods I've so far ...
2
votes
0answers
15 views

Proof that maximal interval of existence exist and bounded

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
1
vote
0answers
36 views

Proving that maximal interval of existence exists and that solution is unque

For each $\lambda\in \mathbb{R}$, let $\varphi_{\lambda}$ : $J_{\lambda}\rightarrow \mathbb{R}$ denote the solution to the following initial value problem: $$ ...
2
votes
4answers
65 views

Second order homogenous non-linear DE: $3(x')^2 - 2x''x=0$

How do I solve this for $x$? $$3\dot{x}^2-2\ddot{x}x=0$$ $$\Leftrightarrow$$ $$3(x')^2 - 2x''x=0 $$ Note: This comes from my working here(on stack exchange meta sandbox[newest activity]) List of ...
5
votes
2answers
94 views

Green's function for $y''+y=f(x)$

This example is taken from the Wikipedia's article. Namely, find the Green's function for $$y'' + y = f(x)$$ with boundary conditions: $$y(0) = y(\frac {\pi} {2}) = 0.$$ The defining equation for ...
0
votes
2answers
31 views

Two very similar solutions to a differential equation through two different methods

In our differential equation class, we learned of two methods to solve elementary differential equations: integration factors and seperation. We had to solve the differential equation (k is a ...
0
votes
0answers
36 views

What is the Riemann surface of the exponential integral?

I have recently encountered a differential equation whose solution has a term \begin{equation} \frac{1}{2}e^{-\frac{1}{2 \varepsilon} e^{i \tau}} \int_{\tau_0}^\tau e^{\frac{1}{2 \varepsilon} e^{i ...
1
vote
1answer
38 views

Integrate and derivative

i'm not able to explain the following step: $\frac{1}{k+v(x)}=\frac{d^2 v}{dx^2}$ by integrating this equation: $(C-\frac{1}{k+v(x)})^{\frac{1}{2}}=\frac{dv}{dx}$ Please, if somebody can help i'll ...
8
votes
3answers
202 views

Integrating $d\psi=(x+y)dx +x_0dy$

I am quite embarrassed to ask this question, as I know i have lost track of the concept here, but Ill nevertheless ask it. I was going through Mathematical methods for physicists, and there was an ...
0
votes
0answers
36 views

Integration of a differential equation

I've got some problems with integrating a ODE, so maybe someone could add some words of advice. Given the following equation: $z''(x)-2\gamma z'(x) +p(x)z(x)=0$, $(1)$ and $\varphi(x)=z(x)e^{-\gamma ...
1
vote
2answers
58 views

How to prove that a derivative of a formula equals to another formula.

If $u= \ln(\tan x+\tan y+\tan z)$ prove $$\sin 2x \dfrac{du}{dx} + \sin 2y \dfrac{du}{dy} + \sin 2z \dfrac{du}{dz}=2 $$ My answwer was like this: $$u' =\dfrac{ 1}{\tan x+\tan y+\tan z} \cdot( ...
1
vote
2answers
43 views

Integrable combinations - I can't seem to arrive at the given answer

I need help! I can't seem to arrive at the answer given in our textbook. I'm new here, so I really need help. The instruction says that I need to solve this D.E by recognizing integrable ...
2
votes
1answer
154 views

How do I Solve This Kind of Differential Equation? [closed]

How do I solve this differential equation? $$y(2x+y^2)dx+x(y^2-x)dy=0$$
0
votes
1answer
23 views

Laplace transform on a non-standard sort of problem

I don't know where a laplace comes into play here: $\ddot{a}+2a=0,a(0)=b_1,\dot{a}(0)=b_2$ I am meant to solve the above using a Laplace transform, but I don't see how I would use it here? I ...
0
votes
0answers
26 views

Derivative with respect to a function

We have a function ${f(s,{\psi(s)}_{3\times 1})}_{3\times1}\tag1$ Given Data $f,\psi$ are matrices and their dimensions are already given in the question s is not a matrix, it is a scalar ...
0
votes
1answer
31 views

Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
1
vote
1answer
52 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
1
vote
1answer
34 views

Solving second order differential equation numerically with values given at intermediate points.

I need to numerically solve the equation, \begin{equation} y''(x) + p(x)y(x) = 1 \end{equation} in the range [a,b] with conditions \begin{eqnarray} y'(\alpha) &=& 1\\ y(\beta) &=& 0 ...
0
votes
5answers
111 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
1
vote
1answer
58 views

System of ODE - Solution

I have a system of ODE to solve $$ A_{5 \times 5}\ddot{q}(t)_{5 \times 1}+ B_{5 \times 5}\dot{q}(t)_{5 \times 1}+ C_{5 \times 1} =0\tag 1$$ Given Data $A,B,C$ are constants.We know what is ...
6
votes
1answer
214 views

Integration of combination of Bessel Function and Exponential Function

I have read "Watson:Treatise Theory of Bessel Function", "Table of Integration, Series and Product", "Handbook of Mathematical Functions, Formulas, Graphs and Mathematical Tables" and other online ...
1
vote
6answers
95 views

Initial value $\left ( \frac{dy}{dt} \right )+3y=11$, $y(0)=1$

I have never done an initial value problem, and would like some help on how to start this please.
1
vote
1answer
59 views

Matrix - Commutative property

I have a rotation matrix represented as $R(t)=e^{B(t)},\tag 1$ where $B(t)$ is a skew symmetric matrix (since any rotation matrix can be expressed as a matrix exponent of a skew symmetric matrix), ...
1
vote
3answers
178 views

Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
1
vote
4answers
50 views

How to solve $(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$

Solve $$(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$$ I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set ...
0
votes
3answers
84 views

Why does solving $\int \frac{v}{9.8-0.0025v^2}\mathrm{d}v=\int1{d}x$ for $v^2$ in terms of $x$ produce 2 completely different answers?

In this question $g=9.8$ (acceleration of free fall). You are also given that when $x=0$ $v=0$. My answer is $v^2=400g(1-e^\frac{x}{200})$. I obtained it by integrating both sides so that ...
4
votes
3answers
118 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
2
votes
1answer
73 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
0
votes
0answers
24 views

Integration of nonlinear and linear ODEs

\begin{equation} \frac{dc_1}{d\tau}= \alpha I(1-c_{0}) + c_{1} (-K_{F} - K_{D}-K_{N} s_{0}-K_{P}(1-q_{0}))+ c_{0}(-K_{N} s_{1}+K_{P}q_{1}), \nonumber \end{equation} \begin{equation} ...
3
votes
4answers
85 views

Differential equation which has following solution $y=\frac{1}{1+\exp(ax)}$

Is there any linear differential equation which has following solution $$y=\frac{1}{1+\exp(ax)}$$ $a$ is constant. something like: $$ y'' + by' +cy + \alpha = 0$$ where $b$, $\alpha$ and $c$ are ...
2
votes
1answer
35 views

Solve 2 connected ODEs describing a domain

This problem confused me for a long time. I have 2 ODEs which describe part of our domain. They are connected at middle: $$ \frac{d^2}{dx^2} u = -a, x<x_0 $$ $$ \frac{d^2}{dx^2} u - \frac{u}{b^2}= ...
1
vote
4answers
68 views

Differential equation with the solution of $(1+ax/2)\exp(-ax)$

Is there any linear differential equation which has following solution $$y=(1+ax/2)\exp(-ax)$$ $a$ is constant.
0
votes
1answer
71 views

Two methods of finding a function $f$ such that $Mdx+Ndy=0$ on the curves $f(x,y)=c$

this problem is from my class,i did one way and got one answer,professor did it in another way and got another answer.question is:Find $f(x,y)=constant$ where differential equation is ...