Differential equation Definition and 1000 Threads

We choose an approximative solution given by $$ u_N(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \cos nx + b_n \sin nx $$ Comparing this approximative solution with the differential equation yields that $$ \frac{a_0}{2} = a $$ and the boundary conditions yields the equation system $$ a + \sum_{n=1}^N...

One thing that is given in paper (attached) is a operating set point for temperature which is given as 20 for day and 16 for night but I do not know whether its initial condition for temperature or not. Can anyone please guide me that what kind of equation is it and how can I solve it with these...

$$p=\gamma m v$$ $$F = \frac {md (\gamma v}{dt}$$ $$\int{F dt} = \int{md (\gamma v}$$ $$F t= \gamma mv$$ At this step, I don't know how to make v as explicit function of t, since gamma is a function of v too. Thankss

Here is my attempt at a solution: y = f(x) yp - ym = dy/dx(xp-xm) ym = 0 yp = dy/dx(xp-xm) xm=ypdy/dx + xm xm is midpoint of OT xm = (ypdy/dx + xm) /2 Not sure where to go from there because the solution from the link uses with the midpoint of the points A and B intersecting the x-axis...

I have a problem. The task is to develop an differential equation of the airflow of a balloon. I know that it is dependent on the volume and pressure. But I can't get a good differential equasion. Can someone help me? [Thread moved from the technical forums, so no Homework Help Template is shown]

Hi, I really struggled to dig valuable things out of internet and books related to high order homogeneous differential equation with variable coefficients but I have nothing. All methods I see involves given solution and try to find others(like reduction of order method), even for second order...

This is quite literally a showerthought; a differential equation is a statement that holds for all ##x## within a specified domain, e.g. ##f''(x) + 5f'(x) + 6f(x) = 0##. So why is it called a differential equation, and not a differential identity? Perhaps because it only holds for a specific set...

I am unable to complete the first part of the question. After I plug in the function for psi into the differential equation I am stuck: $$\frac {d \psi (r)}{dr} = -\frac 1 a_0 \psi (r), \frac d{dr} \biggl(r^2 \frac {d\psi (r)}{dr} \biggr) = -\frac 1 {a_0}\frac d {dr} \bigl[r^2 \psi(r) \bigr] =...

I'm trying to solve a differential equation of the form $$\frac{A'(x)}{A(x)}f(x,y) = \frac{B'(y)}{B(y)}$$ where prime denotes differentiation. I know that for the case ##f(x,y) = \text{constant}## we just equal each side to a same constant. Can I do that also for the case where ##f(x,y)## is not...

How does one solve the partial differential equation Uxx+Uyy+Uzz=C when C is non-zero. Here U is a function of x,y and z where (x,y,z) lies in the ball centered at 0 of radius 1 and U=0 on the boundary. Uxx, Uyy and Uzz denote second partial derivatives with respect to x, y and z. Any hints on...

Suppose we displace the pendulum bob ##A## an angle ##\theta_0## initially, and let go. This is equivalent to giving it an initial horizontal displacement of ##X## and an initial vertical displacement of ##Y##. Let ##Y## initially be a negative number, and ##X## initially be positive. I observe...

I let ##M = 4xy + 1## and ##N = 2x^2 + \cos{(y)}##. Since ##\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}##, the equation is exact and we have $$\frac{\partial f(x,y)}{\partial x} = 4xy + 1$$ From inspection, you can tell this has to lead to $$f(x,y) = 2x^2 y + x + h(y)$$ and we...

The formula for general oscillation is: The formula for underdamping oscillation is: where λ = -γ +- sqart(γ^2 - ω^2), whereas A+ and A- , as well as λ+ and λ-, are complex conjugates of each other. After some operations, we get: x(t) = Ae^(-γx)[e^i(θ+ωx) +e^-i(θ+ωx)], where A is the modulus...

Let $f:\mathbb R\to \mathbb R$ be a twice-differentiable function such that $f(x)+f^{\prime\prime}(x)=-x|\sin(x)|f'(x)$ for $x\geq 0$. Assume that $f(0)=-3$ and $f'(0)=4$. Then what is the maximum value that $f$ achieves on the positive real line? a) 4 b) 3 c) 5 d) Maximum value does not exist...

I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...

I need to solve ∂2Φ/∂s2 + (1/s)*∂Φ/ds - C = 0 Where s is a radial coordinate and C is a constant. I know this is fairly simple but I haven't had to solve a problem like this in a long time. Can someone advise me on how to begin working towards a general solution? Is the method of...

From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...

253 Which of the following is the solution to the differential equation condition $$\dfrac{dy}{dx}=2\sin x$$ with the initial condition $$y(\pi)=1$$ a. $y=2\cos{x}+3$ b. $y=2\cos{x}-1$ c. $y=-2\cos{x}+3$ d. $y=-2\cos{x}+1$ e. $y=-2\cos{x}-1$ integrate $y=\displaystyle\int 2\sin...

I tried to derive this by myself but I'm stuck. What i did it to substitute a_{1} with a_{1} +\Delta a_{1} in the first equation, getting: (a_{1}+\Delta a_{1})\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t) and trying to subtract a_{1}\dot{y}(t)+y(t)=b_{0}u(t)+b_{1}\dot{u}(t) to it. But it's not...

I'm having a hard time grasping the concept of reducing the two recursive relations at the end of the frobenius method. For example, 2xy''+y'+y=0 after going through all the math i get y1(x) = C1[1-x+1/6*x^2-1/90*x^3+...] y2(x) = C2x^1/2[1-1/3*x+1/30*x^2-1/630*x^3+...] I know those are right...

I wish to know if there is a method to work out x(t). [No matter which form f(t) is] Thank you~

on introducing a term on both sides, we have ##(x^2+xy-2xy)y^{'}=x^2+y^2-2xy## ##(x^2-xy)y^{'}=(x-y)^2## ##x(x-y)y^{'}=(x-y)^2## ##xy^{'}=(x-y)## ##y^{'}=1-y/x## ## v+x v^{'}=1-v## ...ok are the steps correct before i continue?

Hi, Could you please help me to solve a second-order differential equation given below ∂M/r∂r+∂2M/∂r2 = A [Moderator's note: Moved from a technical forum and thus no template.]

Homework Statement: first order non linear equation Homework Equations: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4 a,b,z,v,u are constant t0=0 , T=T0 Hi, i need find an experession of T as function of t from this first order nonlinear equation: dT/dt=a-bT-Z[1/(1+vt)^2]-uT^4 a,b,z,v,u are constant...

1.) Laplace transform of differential equation, where L is the Laplace transform of y: s2L - sy(0) - y'(0) + 9L = -3e-πs/2 = s2L - s+ 9L = -3e-πs/2 2.) Solve for L L = (-3e-πs/2 + s) / (s2 + 9) 3.) Solve for y by performing the inverse Laplace on L Decompose L into 2 parts: L =...

So this is what I have done: ##f'(t)=k*f(t)*(A-f(t))*(1-sin(\frac{pi*x}{12}))## ##\frac{1}{f(t)*(A-f(t))}=k*(1-sin(\frac{pi*x}{12}))## I see that the left can be written as this (using partial fractions): ##1/A(\frac{1}{f(t)}-\frac{1}{A-f(t)})## And then I take the integral on both sides and...

If we take a flat universe dominated by radiation, the scale factor is ##a(t)=t^{1/2}## which can be derived from the first Friedmann Equation:$$(\dot a/a)^2 = \frac{8\pi G}{3c^2}\varepsilon(t)-\frac{kc^2}{R_0^2 a(t)^2}$$ But suppose I want to show this using the second Friedmann Equation (Also...

Consider the following linear first-order PDE, Find the solution φ(x,y) by choosing a suitable boundary condition for the case f(x,y)=y and g(x,y)=x. --------------------------------------------------------------------------- The equation above is the PDE I have to solve and I denoted the...

Introducing the new variables ##u## and ##v##, the chain rule gives ##\dfrac{{\partial{f}}}{{\partial{x}}}=\dfrac{{\partial{f}}}{{\partial{u}}} \dfrac{{\partial{u}}}{{\partial{x}}}+\dfrac{{\partial{f}}}{{\partial{v}}} \dfrac{{\partial{v}}}{{\partial{x}}}##...

i am doing research to make criteria by which i can identify easily linear and non-linear and also identify its singular or not by doing simple test.please help me in this regard.

When reading through Shankar's Principles of Quantum Mechanics, I came across this ODE \psi''-y^2\psi=0 solved in the limit where y tends to infinity. I have tried separating variables and attempted to use an integrating factor to solve this in the general case before taking the limit, but...

I tried the substitution ##y=e^{\int z(x)}##,##z(x)## is an arbitrary function to be determined. Substitute this to the original differential equation,and dividing ##y^2## yields ##(z+1)z'+z^3+z^2+z=0##,which is a first order differential equation. Trying to solve this first order differential...

Hi all. I have another exam question that I am not so sure about. I've solved similar problems in textbooks but I have a feeling once again that the correct way to solve this problem is much simpler and eluding me. Especially because my answer to a) is already the solution to c) and d) (I did...

The ODE is: \begin{equation} (y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0 \end{equation} Where y(x) and z(x) are real unknown functions of x, m is a constant. I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...

This is my attempt: \frac{dy}{dx} = \frac{y^2 - 1}{x^2 - 1} \\ \int \frac{dy}{y^2 - 1} = \int \frac{dx}{x^2 - 1} \\ \ln \left| \frac{y-1}{y+1} \right| + C_1 = \ln \left| \frac{x-1}{x+1} \right| + C_2 \\ \ln \left| \frac{y-1}{y+1} \right| = \ln \left| \frac{x-1}{x+1} \right| + C Since y(2) =...

Here, M = ##siny*cosy +xcos^{2}y ## and N = x ## M_y = (1/2)cos(2y) -xsin(2y)## and ##N_x = 1## Theorems: If R = ## \frac{1}{N} (M_y - N_x) = f(x), then I.F. = e^{ \int f(x) dx} ## If R = ## \frac{1}{M} (N_x - M_y) = g(y), then I.F. = e^{ \int g(x) dx} ## Neither is holding true. What should...

Hi, I was trying to see if the following differential equation could be solved using Laplace transform; its solution is y=x^4/16. You can see below that I'm not able to proceed because I don't know the Laplace pair of xy^(1/2). Is it possible to solve the above equation using Laplace...

ov!347 nmh{1000} Convert the differential equation $$y''+5y'+6y=e^x$$ into a system of first order (nonhomogeneous) differential equations and solve the system. the characteristic equation is $$\lambda^2+5\lambda+6=e^x$$ factor $$(\lambda+2)(\lambda+3)=e^x$$ ok not real sure what to do with...

ln(y) +y = ln(x) + x +C y=?

dM/dY = x+2y+1 dN/dx = 1 (My-Nx)/n = 1 Integrating Factor => e^∫1dx= e^x (xye^x+ye^x+ye^x)dx + (xe^x+2ye^x)dy = 0 dM/dY =xye^x+e^x+2ye^x dN/dx = xye^x+e^x+2ye^x Exact ∫dF/dy * dy = ∫ (xe^x+2ye^x)dy F = xy*e^x + y^2*e^x + c(x) dF/dx = xy*e^x + y*e^x + y^2 * e^x + c'(x)...

So in my previous math class I spotted on my book an exercise that I couldn't solve. We had to find the general solution for the differential equation. This was the exercise: 4y'' - 4y' + y = ex/2√(1-x2) Can anyone tell me how to solve this step by step?

. Above is the figure of the problem. I am trying to solve x(t) and differentiate it to obtain v(t); however, I have difficulty solving the differential equation shown below. $$ v(t)=\int a(t)dt=\int \frac{B(\varepsilon-Blv)d}{Rm}dt \Rightarrow \frac{dx}{dt}=\frac{B\varepsilon...

The rate at which glucose enters the bloodstream is ##r## units per minute so: ## \frac{dI}{dt} = r ## The rate at which it leaves the body is: ##\frac {dE}{dt} = -k Q(t) ## Then the rate at which the glucose in the body changes is: A) ## Q'(t) = \frac{dI}{dt} + \frac {dE}{dt} = r - k...

How do I start this? I plugged the differential equation at wolfram alpha and it semmed too complicated for such an exercise. I've also looked at a sample of an answer on cheeg where the initial approach is to rewrite the equation as ##\frac{d}{dt} (\frac{\dot\theta^2}{2}-cos(\theta)) = 0## How...

Hello, I have to solve this second order differential equation. It's like a string vibrating equation but with a constant c: $$\frac{{\partial^2 u}}{{\partial t^2}}=k\frac{{\partial^2 u}}{{\partial x^2}}+c$$ B.C $$u(0,t)=0$$ $$u(1,t)=2c_0$$ c_0 is also a constant I.C $$u(x,0)=c_0(1-\cos\pi...

Hey all, it's been awhile since done any calculus or DE's but was trying out some modelling (best price price per item for bulk value deals as a function of time and amount), in the last line i have f(n,t) implicitly. Any pointers or techniques for solving such things?