Differential equations problems, solved and explained

Recent questions in Differential equations

2022-05-09

estroishave4 2022-05-03 Answered

Series solution of $x y^{^{″}} + 2 y^{^{'}} - x y = 0$
I get $r (r + 1) = 0, (r + 1) (r + 2) c_{1} = 0$ and
$c_{n + 1} = \frac{c_{n - 1}}{(n + 1 + r) (n + r + 2)}$
The first equation gives the indicial roots $r = - 1$ and $r = 0$ . The case for $r = 0$ is fine.
For $r = - 1$ , I don't see how $c_{1} = 0$ is implied by the second equation. The way I see it, since $1 + r = 0$ when $r = - 1, c_{1}$ is not necessarily zero here, but this leads to a different solution to what is in the text book. What am I missing? Why is $c_{1}$ forced to be zero?

sexe4yo 2022-05-02 Answered

Second Order Differential Equations $a y^{″} + b y^{'} + c y = 0$ , without complex numbers

hapantad2j 2022-05-02 Answered

Question in population dynamics using exponential growth rate equation
Given population doubles in 20 minutes, what is intrinsic growth rate r?
Attempt: Given population doubles, using exponential growth rate we have $\frac{d N}{d t} = 2 N$ so $N (t) = N_{0} e^{2 t}$ therefore $r = 2$ , but I have a feeling this is wrong since 20 minutes should be used somewhere around here.

Magdalena Norton 2022-05-02 Answered

$\frac{d y}{d x} = y + 1$ Solving the above given differential equation, yields the following general solution. $y + 1 = e^{x + C}$
$y = C e^{x} - 1$
$\Rightarrow$ Solution $y = - 1$ at $C = 0$
Can I say that $y = - 1$ is a singular solution?

Sydney Stanley 2022-05-02 Answered

$(1 - x^{2}) y^{″} - x y^{'} + m^{2} y = 0$ , m is a constant. I have to show that $m \in Z \Rightarrow d e g (y) = m$ such that y is a solution of the ode.

yert21trey123z7 2022-05-01 Answered

Second order differential inequality and comparison theorem.
If $x : [0, 1] \Rightarrow R, x \in C^{\infty}$ that satisfies
${\begin{cases} \ddot{x} \leq - 2 x \\ x (0) = x (1) = 0 \end{cases}$
is it true that $x \geq 0 on [0, 1]$ ?
I observed that if $x \geq 0$ , then $\ddot{x} \leq 0$ , so x is concave. However I could not find out any other properties. I think if there exists a solution of the following differential equation
${\begin{cases} \ddot{x} = - 2 x \\ x (0) = x (1) = 0 \end{cases}$
then some kind of comparative theorem can be used. However, there is no non-trivial solution for this.
If the above proposition is incorrect, could you give me a counterexample?

hadaasyj 2022-05-01 Answered

Prove instability using Lyapunov function
$x^{'} = x^{3} + x y$
$y^{'} = - y + y^{2} + x y - x^{3}$

pacificak8m 2022-05-01 Answered

Linearization around the equilibrium point for a system of differential equations
$\frac{d N}{d t} = μ N (1 - \frac{N}{K}) + N \int_{0}^{\infty} p (a, t) d a$
$\frac{\partial p}{\partial t} + \frac{\partial p}{\partial a} = k p (a, t) .$
Linearizing it around $(N = K, p = 0)$ , I am getting the system
$\frac{d N}{d t} = h - μ N + N \int_{0}^{\infty} p (a, t) d a$
$\frac{\partial p}{\partial t} + \frac{\partial p}{\partial a} = k p (a, t),$
where $h = μ K$ .
The reason I am getting this system is because I am considering $N \int_{0}^{\infty} p (a, t) d a$ as linear term. Is this linearisation right or the correct system is $\frac{d N}{d t} = h - μ N$
$\frac{\partial p}{\partial t} + \frac{\partial p}{\partial a} = k p (a, t),$
where $h = μ K$ .

Ali Marshall 2022-05-01 Answered

$x^{2} y^{″} + (3 x^{2} + 4 x) y^{'} + 2 (x^{2} + 3 x + 1) y = 0$
I try to solve this equation by finding $μ$ such that the equation become exact.( I know there are other ways for solving this equation).
${(μ y^{'})}^{'} = μ^{'} y^{'} + μ y^{″} = μ y^{″} + μ \frac{3 x^{2} + 4 x}{x^{2}} y^{'} + μ \frac{2 (x^{2} + 3 x + 1)}{x^{2}} y \Rightarrow$
$μ^{'} y^{'} = μ \frac{3 x^{2} + 4 x}{x^{2}} y^{'} + μ \frac{2 (x^{2} + 3 x + 1)}{x^{2}} y \Rightarrow \frac{μ^{'}}{μ} = \frac{3 x^{2} + 4 x}{x^{2}} + \frac{2 (x^{2} + 3 x + 1)}{x^{2}} \frac{y}{y^{'}}$
How am I supposed to solve it?

iyiswad9k 2022-05-01 Answered

Proving single solution to an initial value problem
$y' = | \frac{1}{1 + x^{2}} + \sin | x^{2} + \arctan y^{2} | |, y (x_{0}) = y_{0}$
or each $(x_{0}, y_{0}) \in R \times R$ I need to prove that there is a single solution defined on $R$

Libby Boone 2022-05-01 Answered

Proving single solution of initial value problem is increasing
Given the initial value problem
$y^{'} (x) = y (x) - \sin {y (x)}, y (0) = 1$

Kaiya Hardin 2022-04-30 Answered

Second Order Nonhomogeneous Differential Equation (Method of Undetermined Coefficients)
Find the general solution of the following Differential equation $y^{″} - 2 y^{'} + 10 y = e^{x} \cos (3 x)$ . We know that the general solution for 2nd order Nonhomogeneous differential equations is the sum of $y_{p} + y_{c}$ where $y_{c}$ is the general solution of the homogeneous equation and $y_{p}$ the solution of the nonhomogeneous. Therefore $y_{c} = e^{x} (c_{1} \cos (3 x) + c_{2} \cos (3 x))$ . Now we have to find $y_{p}$ . I know in fact that $y_{c} = e^{x} (c_{1} \cos (3 x) + c_{2} \cos (3 x))$ . Now we have to find yp. I know in fact that $y_{p} = e^{\times} \frac{\sin (3 x)}{6}$ but i do not know how to get there.

adiadas8o7 2022-04-30 Answered

Second-order ODE involving two functions
I am wondering how to find a general analytical solution to the following ODE:
$\frac{d y}{d t} \frac{d^{2} x}{{d t}^{2}} = \frac{d x}{d t} \frac{d^{2} y}{{d t}^{2}}$
The solution method might be relatively simple; but right now I don't know how to approach this problem.

micelije1mw 2022-04-30 Answered

Removal of absolute signs
An object is dropped from a cliff. The object leaves with zero speed, and t seconds later its speed v metres per second satisfies the differential equation
$\frac{d v}{d t} = 10 - 0.1 v^{2}$
So I found t in terms of v
$t = \frac{1}{2} \ln | \frac{10 + v}{10 - v} |$
The questions goes on like this: Find the speed of the object after 1 second. Part of the answer key shows this
$t = \frac{1}{2} \ln | \frac{10 + v}{10 - v} |$
$2 t = \ln | \frac{10 + v}{10 - v} |$
$e^{2 t} = \frac{10 + v}{10 - v}$
So here's my question: why is it not like this?
$\pm e^{2 t} = \frac{10 + v}{10 - v}$
Why can you ignore the absolute sign?

Magdalena Norton 2022-04-30 Answered

Question on a differential equation
Let $x : R \Rightarrow R$ a solution of the differenzial equation:
$5 x^{″} (t) + 10 x^{'} (t) + 6 x (t) = 0$
Proof that the function:
$f : ℝ \Rightarrow ℝ$

Maurice Maldonado 2022-04-30 Answered

How to solve this nonlinear diff eq of celestial mechanics?
${(\dot{r})}^{2} = \frac{2 μ}{r} + 2 h$
Where mu and h are constants.
I have no idea how to solve it, maybe there is a trick I didn't know.
The only thing that came in mind is to integrate
$\int \frac{d r}{\sqrt{\frac{2 μ}{r} + 2 h}} = \int d t$
but I don't think this is really a solution, since I don't know too how to evaluate the integral in terms of elementary functions.

Elise Winters 2022-04-29 Answered

Let u be a solution of the differential equation $y^{'} + x y = 0$ and $ϕ = u ψ$ be a solution of the differential equation $y^{″} + 2 x y^{'} + (x^{2} + 2) y = 0$ such that $ϕ (0) = 1, ϕ^{'} (0) = 0$ . Then $ϕ (x)$ is equal to
A) $(\cos^{2} (x)) e^{\frac{- x^{2}}{2}}$
B) $(\cos (x)) e^{\frac{- x^{2}}{2}}$
C) $(1 + x^{2}) e^{\frac{- x^{2}}{2}}$
D) $(\cos (x)) e^{- x^{2}}$

Alejandro Atkins 2022-04-29 Answered

Let $f : R \Rightarrow R$ be a differentiable function, and suppose $f = f^{'}$ and $f (0) = 1$ . Then prove $f (x) \neq 0$ for all $x \in R$

wuntsongo0cy 2022-04-29 Answered

Let N be a positive integer. Find all real numbers a such that the differential equation $\frac{d^{2} y}{d x^{2}} - 4 a \frac{d y}{d x} + 3 y = 0$ has a nontrivial solution satisfying the conditions $y (0) = 0$ and $y (2 N π) = 0$

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