Differential Equations and Linear Algebra - Math Questions and Answers

Recent questions in Differential equations

Aryan Salinas 2022-03-24

How do I compute the maximum value of the solution of this differential equation?
${\begin{aligned} \frac{d y}{d t} & = f (y) = 1 + k ϵ - k y - e^{- y} \\ y (0) & = ϵ \end{aligned}$

alparcero97oy 2022-03-24

Help to find the solution of
$y^{'} = (\begin{matrix} - 2 & 1 \\ 1 & - 2 \end{matrix}) \cdot y + (\begin{matrix} 2 \sin (t) \\ 2 (\cos (t) - \sin (t)) \end{matrix})$

Nancy Richmond 2022-03-24

Help needed solving logistic differential equation with initial conditions
$\frac{d P}{d t} = P (10 - 2 P)$
with initial conditions of $P (0) = 1 .$

k4s3thg368 2022-03-24

Given: $y^{2} = c e^{x} - x - 1$
Find the orthogonal family of curves and write 2 curves from both families that pass in (2,0).

ikramkeyslo4s 2022-03-24

Limit of a separable equation:
Consider the equation
$\frac{d y}{d t} = k (a - y) (b - y)$
where a,b and k are constants. Assuming $y (0) = 0$
a) Solve for y(t) when $a = b$
b) Solve for the case $0 < a < b$
c) By considering the limit $b \to a$ in (b) show that the two results are consistent

Jazmyn Holden 2022-03-23

Solve the heat equation using a transform method
$k \frac{\partial^{2} U}{\partial x^{2}} = \frac{\partial U}{\partial t}$
subject to
$U (0, t) = 1, t > 0$
$U (x, 0) = e^{- x}, x > 0$

Addison Fuller 2022-03-23

Solve $3 y^{'} + 6 x y = 6 e^{- x}$ where $y (0) = 1$ .
I am trying to solve $3 y^{'} + 6 x y = 6 e^{- x}$ where $y (0) = 1$ . This is a first-order linear equation. Dividing by 3 and multiplying by $e^{x^{2}}$ , we have
$e^{x^{2}} y^{'} + 2 x e^{x^{2}} y = 2 e^{x^{2} - x} .$
Thus, we have $\frac{d}{d x} (e^{x^{2}} y) = 2 e^{x^{2} - x}$ .
Now, we have $e^{x^{2}} y = 2 \int e^{x^{2} - x} d x$ .
How do I calculate this integral? The solution is supposedly $y (x) = (2 x + 1) e^{- x^{2}}$ .

resaplaha7dl 2022-03-23

Limit for $t r i g h t a r r w o \infty$ of a solution to an ODE
Consider the Cauchy problem:
${\begin{cases} y^{'} (t) = \frac{t^{2}}{2 + \sin (t^{2})} c o s (y (t)^{2}) \\ y (0) = 0 \end{cases}$

ezpimpin6988ok1n 2022-03-23

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

meli199939f 2022-03-23

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 $y (2 N π) = 0$

Nadia Clayton 2022-03-22

Solve this nonhomegenous ode
$y^{″} + 4 y = \cos (2 x)$

juctommaccedo662f 2022-03-22

Solve this equation in Sturm-Liouville.
$(1 - x^{2}) y^{″} - x y^{'} + λ y = 0$
My solution:
$μ (1 - x^{2}) y^{″} - μ x y^{'} + μ λ y = {(p y^{'})}^{'} + (λ r - q) y$
$μ (1 - x^{2}) y^{″} - μ x y^{'} + μ λ y = p^{'} y^{'} + p y^{″} + (λ r - q) y$
$p^{'} = μ x, p = μ (1 - x^{2}) \Rightarrow \frac{p^{'}}{p} = \frac{x}{1 - x^{2}} \Rightarrow \ln (p) = - \frac{1}{2} \ln | 1 - x^{2} | \Rightarrow p = \frac{1}{\sqrt{(| 1 - x^{2} |)}}$
Then $μ = \frac{1}{{(1 - x^{2})}^{3}}$
$\frac{1}{{(1 - x^{2})}^{2}} y^{″} - \frac{1}{{(1 - x^{2})}^{3}} x y^{'} + \frac{1}{{(1 - x^{2})}^{3}} λ y = p^{'} y^{'} + p y^{″} + (λ r - q) y$

parksinta8rkv 2022-03-22

Solve the differential equation:
$\frac{d}{d x} (2 y y^{'}) = {(y^{'})}^{2}$

Petrolovujhm 2022-03-22

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 $h = μ K$ .

Beryneingmk39 2022-03-22

Function whose minimum is independent of the input?
I have a question that asks me to show that the functional
$I (x) = \int_{{x_{1}}}^{x_{2}} (x^{2} + 3 y^{2}) y^{'} + 2 x y d x$
has an extremal that is independent of the choice of y that joins two arbitrary points $(x_{1}, y_{1}), (x_{2}, y_{2})$ .
So I tried solving for the extremal, which give me the euler lagrange equation:
$6 y y^{'} + 2 x = 2 x + 6 y y^{'}$
That's not an error it seems that $\frac{d}{d x} (\frac{\partial F}{\partial y^{'}}) = \frac{\partial F}{\partial y}$ .
Ok then no wonder that the path doesn't matter. But now I am trying to compute the value of the functional for the two points, I tried a straight line between the 2, but this is giving me a messy expression, it's computable but ugly:
$J (y) = \int_{{x_{1}}}^{x_{2}} s (x^{2} + 3 {(s x + k)}^{2}) + 2 x (s x + k) d x$
(where s and k are the linear coefficients of the line connecting the two points).

siliciooy0j 2022-03-21

Solve the equation:
$x (2 y + 1) d x = y (x^{2} - 3 x + 2) d y$

Zack Mora 2022-03-21

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) $(\cos (x)) e^{- x^{2}}$

tibukooinm 2022-03-21

Let $f \in C_{o}^{\infty} (R^{n})$ . Propose formulas of the form $u (x) = d i s p l a y s t y \leq \int_{{}^{{} \frac{\hat{f} (δ) e^{i x δ}}{p (δ)} d δ$
to solve:
i) $\sum_{j = 1}^{n} \frac{\partial^{4} u (x)}{\partial x_{j}^{4}} + u (x) = f (x)$
andii) $\sum_{k, l}^{{} \frac{\partial^{4} u (x)}{\partial x_{k}^{2} \partial x_{l}^{2}} - 2 \cdot d i s p l a y s t y \leq \sum_{k = 1}^{n} \frac{\partial^{2} u (x)}{\partial x_{k}^{2}} + u (x) = f (x)$

sempteim245 2022-03-20

Solve the ode using fixed-point iteration
$- {[(1 + u^{4}) u_{x}]}_{x} = \sin (x) + \sin (5 x)$ , where the domain is $[0, 2 π] u (0) = u (2 π) = 0$ for boundaries.

Pablo Dennis 2022-03-20

Solve the following differential equation by the form of homogeneous equation. Letting $y = v x$
The equation: $x^{2} \frac{d y}{d x} + x y + 1 = 0$

One of the most challenging parts of Calculus 2 relates to differential equations. It’s recommended to start with the ordinary differential equations as these will help you understand the concepts. If you are not sure how to work with variables and functions, start with at least one equation that is provided in differential equations examples. See the unknown function and seek the answers by comparing what you have available. The majority of differential equations problems will provide you with the basics. If you are not sure how to find differential equations solutions, think about the relationship between functions and derivatives.

Calculus 2

Differential Equations and Linear Algebra Q&A