Articles of systems of equations

Finding the kernel of a linear map

Our exercise is to find all solutions to the equation $Ax = 0$, among others for the following matrix $$A =\begin{pmatrix} 6 & 3 & -9 \\ 2 & 1 & -3 \\ -4 & -2 & 6 \end{pmatrix}.$$ This amounts to finding the kernel, and obviously, the rows of the matrix are multiples of […]

Easiest way to solve system of linear equations involving singular matrix

I am trying to balance an unbalanced chemical equation by using setting up a system of linear equations to solve for the stoichiometric coefficients in the chemical equation. After setting up a matrix, to try and solve the system, i cant because one of the matrices is a singular array. I have taken a look […]

Describe all integers a for which the following system of congruences (with one unknown x) has integer solutions:

$$x\equiv a \pmod {100}$$ $$x\equiv a^2 \pmod {35}$$ $$x\equiv 3a-2 \pmod {49}$$ I’m trying to solve this system of congruences, but I’m only familiar with a method for solving when the mods are pairwise coprime. I feel like there ought to be some simplification I can make to reduce it to a system where they […]

Solving the system $(18xy^2+x^3, 27x^2y+54y^3)=(12, 38)$

While answering this question, I got myself stumped with this crazy system with an evil graph: $$\begin{cases} 18xy^2+x^3=12 \\ 27x^2y+54y^3=38 \end{cases}$$ and I wonder whether there is some slick method to find the only real root $(x, y)=(2, 1/3)$ without relying on Cardano’s formula, ideally giving some intuition. This closely reassembles some kind of elliptic […]

Using variation of parameters, how can we assume that nether $y_1$, $y_2$ equal zero?

Here’s the problem I have been given: Use the method of variation of parameters to find a general solution of the following differential equation: $$ a_1y”+a_2y’+a_3y=f(x) $$ And linearly independent solutions $y_1$ and $y_2$ are known. Note: $a_1, a_2,$ and $a_3$ are constants. Assume that $y_p$ is of the form $u_1y_1+u_2y_2$ where $u_1, u_2$ are […]

Finding the general solution to a system of differential equations

How can I solve the following system of differential equations? I am getting confused with the constants of integration… $$\dot{x}=2x-(2+y)e^{y}$$ $$\dot{y}=-y$$ I know that $y=Ce^{-t}$ and the integrating factor method for solving a linear differential equation, but it gets complicated quickly.

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I’m trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = kf_{N-1}-lf_N \end{matrix} \tag{1}$$ with $f_1 (0) = f_0$ and $\forall n \neq 1 \ [f_n(0)=0]$, where $k$, $l$ and $f_0$ are real positive constants. The system may also […]

General solution to a system of non linear equations with a specific pattern

I am seeking a general solution to a system of non linear equations with a specific pattern: Order 1: $$ x_0 = a^2 + b^2 $$ $$ x_1 = 2ab $$ Order 2: $$ x_0 = a^2 + b^2 + c^2 $$ $$ x_1 = 2ab + 2bc $$ $$ x_2 = 2ac $$ Order […]

(Dis)prove that this system has only integral solutions: $\sqrt x+y=7$and $\sqrt y+x=11$

This is the system of equations: $$\sqrt { x } +y=7$$ $$\sqrt { y } +x=11$$ Its pretty visible that the solution is $(x,y)=(9,4)$ For this, I put $x={ p }^{ 2 }$ and $y={ q }^{ 2 }$. Then I subtracted one equation from the another such that I got $4$ on RHS and […]

system of equations $\sqrt{x}+y = 11$ and $x+\sqrt{y} = 7$.

This question already has an answer here: Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$ 7 answers