Differential Equations and Linear Algebra - Math Questions and Answers

Recent questions in Differential equations

Lymnmeatlypamgfm 2022-04-22

Possible Close-form solution for 2 dimensional $\frac{d Σ}{d t} = A Σ + Σ A^{T} + B B^{T}$
When A,B are 2x2 matrix and $\sum (t)$ are PSD, can we expect a close form solution for the following ODE,
$\frac{d Σ}{d t} = A Σ + Σ A^{T} + B B^{T} .$
The problem is from solving the covariance of a time-invariant SDE $x = A x + B d w .$
When x is in two dimensions, I guess there could be a close-form solution for $\sum$ . But I am not sure how to solve it.

hestyllvql 2022-04-22

How am I supposed to find the solution to the non-homogeneous ode ?
${\vec{Y}}^{'} = (\begin{matrix} - 4 & 2 \\ 2 & - 1 \end{matrix}) \vec{Y} + (\begin{matrix} x^{- 1} \\ 2 x^{- 1} + 4 \end{matrix})$

acceloq3c 2022-04-22

Value of differential equation at non existent points in the original function.
$y^{2} + y = x^{3}$

esbagoar7kh 2022-04-22

Solving
$y + x y^{'} = a (1 + x y)$
$y (\frac{1}{a}) = - a$

Esther Hoffman 2022-04-22

Solve linear system of ODE's
It's been years since I formally saw ODE's and I am quite rusty, I don;t remember how to solve linear ODE's.
I have a problem and managed to derive the following system:
${\begin{cases} 2 \ddot{q_{1}} = 2 q_{1} + q_{2} \\ 2 \ddot{q_{2}} = q_{1} \end{cases}$
where $q_{1} and q_{2}$ are scalar functions with parameter t.
It seems to me the solution is a trig function but I am not sure how to go about this problem.

Madilynn Avery 2022-04-21

Nonlinear first oder ODE
I want to solve the following nonlinear ODE
$y^{'} - \frac{y}{x} + \frac{3}{y} (\frac{1}{2} - x) + 1 = 0$ .

Ali Marshall 2022-04-21

Is any direct Adams Moulton formula or how to related those two formula which i am mention.
Are those formulas the same? When I am programming for Adams Moulton fourth order methods I use the 2nd formula which I mentioned in the 2nd image. Is it correct?
If correct, please explain me.
$Y_{n + 1} = Y_{n} + \frac{h}{24} [9 f (x_{n + 1}, y_{n + 1}) + 19 f (x_{n}, y_{n}) - 5 f (x_{n - 11} y_{n - 1}) + f (x_{n - 2}, y_{n - 2})$

Davon Trujillo 2022-04-21

Verify that the following function $y^{2} = e^{(2 x)} + C$ is a solution of the ODE: $y y^{'} = e^{(2 x)}$

Kymani Shepherd 2022-04-21

Verify $y = x \tan x$ is solution to ODE: $x y^{'} = y + x^{2} + y^{2}$
Verify $S y = x \tan {x}$
is a solution to $x y^{'} = y + x^{2} + y^{2}$

Caitlyn Cole 2022-04-21

Variation of parameters method for differential equations.
Change the variable $x = e^{t}$ and then find the general solution for the following differential equation $2 x^{2} y^{″} - 6 x y^{'} + 8 y = 2 x + 2 x^{2} \ln x$
It seems a little suspicious that i can factor out 2 and x first. $x^{2} y^{″} - 3 x y^{'} + 4 y = x (\ln (x) + 1)$ and if we factor out $x^{2}$ now we end up with $y^{″} - \frac{3}{x} y^{'} + \frac{4}{x^{2}} y = \frac{x \ln x + 1}{x}$
(1) Therefore substituting $x = e^{t}$ (1) now becomes $y^{″} - \frac{3}{e^{t}} y^{'} + \frac{4}{e^{2 t}} y = \frac{e^{+} 1}{e^{t}}$ . We need constant coefficients in order to use the variation of parameters method am i right?

Malachi Novak 2022-04-21

Solving 2nd order homogenous linear ODE with a squared coefficient
I am attempting to solve a differential equation of the form:
$\frac{d^{2} f}{d x^{2}} - γ^{2} f = 0$
I have set up and solved the characteristic equation as:
$\begin{aligned} z^{2} - γ^{2} z + 0 z = 0 \\ z (z - γ^{2}) = 0 \end{aligned}$
which is satisfied when $z = 0$ and when $z = γ^{2}$
There are two distinct roots ( $r_{1} and r_{2}$ ) hence the general solution should be of the form
$f (x) = A e^{r_{1} x} + B e^{r_{2} x}$
In this case:
$\begin{aligned} f (x) & = A e^{0 \times x} + B e^{γ^{2} x} \\ = A + B e^{γ^{2} x} \end{aligned}$
Apparently, this solution is incorrect as in the answers it is given as:
$f (x) = A e^{γ x} + B e^{- γ x}$
I would like to know where I went wrong to give me the incorrect solution

Hugh Walls 2022-04-21

Solving a differential equation connecting slope and derivative
$\frac{y (x) - y (a)}{x - a} = y^{'} (x)$

Malachi Novak 2022-04-21

Solving a nonhomogeneous system of equations without matrices
$[\begin{matrix} x^{'} \\ y^{'} \end{matrix}] = [\begin{matrix} 3 & 2 \\ - 2 & - 1 \end{matrix}] [\begin{matrix} x^{'} \\ y^{'} \end{matrix}] + [\begin{matrix} 2 e^{- t} \\ e^{- t} \end{matrix}]$

Davian Lawson 2022-04-21

Solve for f(x).
$\int_{0}^{x^{3} + 1} \frac{f^{'} (t)}{1 + f (t)} d t = \ln (x)$

Libby Boone 2022-04-21

Solutions of the differential equation $x^{2} y^{″} - 4 x y^{'} + 6 y = 0$ .
In one of my test it given to prove that $x^{3} and x^{2} | x |$ are linear independent solutions of the differential equation $x^{2} y^{″} - 4 x y^{'} + 6 y = 0$ on R( here x is independent variable).
But according to me it’s Cauchy Euler equation having general solution as $y = c_{1} x^{3} + c_{2} x^{2}$ , where $c_{1}$ and $c_{2}$ are arbitrary constants. How can be $x^{2} | x |$ a solution of given ODE as I am unable to find its by giving particular values of constants $c_{1} and c_{2}$ ? Please help me to solve it.

Paula Boyer 2022-04-20

How to find the lower and upper bound of a solution of a Cauchy problem?
I am studying some differential equations especially the Cauchy problems, and I encountered this question:
$(E) : {\begin{array}{r} y^{'} = y s i n^{2} (y) \\ y (0) = x_{0} \end{array}$
They supposed that $0 < x_{0} < π$ , and I have to find the upper and lower bounds of the solution, knowing that it is a maximal solution. I don't know where to start. I need some kind of help.

etudiante9c2 2022-04-20

Solving a system of differential equations with the variable x
How to find a basis of solutions of the system for
$y' = (\begin{matrix} 3 x - 1 & x - 1 \\ - x - 2 & x - 2 \end{matrix}) y$
where one solution is $y = (\begin{matrix} y_{1} \\ - y_{1} \end{matrix})$ ?

yopopolin10d 2022-04-20

Solving a differential equation in matrix form but adding a constant
$\begin{aligned} x_{1}^{'} & = 2 x_{1} + 3 x_{2}, \\ x_{2}^{'} & = x_{1} - x_{2} . \end{aligned}$

eszkortwxp 2022-04-20

Solve $(y + 1) d x + (x + 1) d y = 0$

Kason Chang 2022-04-20

Solve ODE $y^{'} (y^{'} + y) = x (x + 1)$
I tried to remove $y^{^{'} 2}$ term by differentiate it wrt x and then replace value in hope that it will turn out some exact form but got stuck after
$2 y^{'} y^{″} - y y^{'} + y^{″} = x - x^{2} + 1$
How i proceed further or my method is wrong ?
Edit: Exact problem
If $y^{'} - x \neq 0$ is a solution of the differential equation
$y^{'} (y^{'} + y) = x (x + 1)$ then y(x) is given by
1. $1 - x - e^{x}$
2. $1 - x - e^{- x}$
3. $1 + x + e^{x}$
4. $1 + x + e^{- x}$

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