What is a solution to the differential equation $y\prime =-{e}^{3x}$?

yamyekay3
2022-09-15
Answered

What is a solution to the differential equation $y\prime =-{e}^{3x}$?

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asked 2020-11-10

Solve differential equation ${y}^{\prime}+y\mathrm{cot}\left(x\right)=\mathrm{sin}\left(2x\right)$

asked 2022-07-12

We have a theorem that says: Let $h:I\to \mathbb{R}$ and $g:J\to \mathbb{R}$ be continuous functions, ${t}_{0}\in I$ and ${y}_{0}\in \text{int(J)}$, hen the differential equation y'(t)=g(y(t))h(t) has a unique solution in some surrounding of ${t}_{0}$ if $g({y}_{0})\ne 0$

How I determine the magnitude of this surrounding? Especially, if I integrate the differential equation and solve it for y(t), am I only able to say: My solution is only unique in the surround of ${y}_{0}$ for which g(y(t)) is not zero, is this the condition that determines my surrounding? I mean, I could have determined a solution that gives me zero for some values of t, but is the only solution for my given problem? Am I correct, that in this case, my theorem is not able to say something about the uniqueness of this solution?

How I determine the magnitude of this surrounding? Especially, if I integrate the differential equation and solve it for y(t), am I only able to say: My solution is only unique in the surround of ${y}_{0}$ for which g(y(t)) is not zero, is this the condition that determines my surrounding? I mean, I could have determined a solution that gives me zero for some values of t, but is the only solution for my given problem? Am I correct, that in this case, my theorem is not able to say something about the uniqueness of this solution?

asked 2021-01-16

Find the general solution of the first-order linear differential equation
$(dy/dx)+(1/x)y=6x+2$ , for x > 0

asked 2022-05-31

I am looking to solve the following equations numerically:

$ax=\frac{d}{dt}(f(x,y,t)\frac{dy}{dt}),\phantom{\rule{1em}{0ex}}by=\frac{d}{dt}(g(x,y,t)\frac{dx}{dt})$

For arbitrary functions f and g and constants a and b. I am struggling to find a way to transform this into a system of first order differential equations that I can pass into a solver. It looks like I will need to define these implicitly, but I'm not sure how to do that.

My best attempt so far is the following:

$\begin{array}{rl}{z}_{1}& =f(x,y,t)\frac{dy}{dt}\\ {z}_{2}& =g(x,y,t)\frac{dx}{dt}\\ {z}_{3}& =ax\\ {z}_{4}& =by\end{array}$

${\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right)}^{\prime}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& \frac{a}{g(t,x,y)}& 0& 0\\ \frac{b}{f(t,x,y)}& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right)$

However, this seems fairly inelegant and assumes that you are always able to divide by f and g. I'm trying to keep this as general as possible, so don't want to make that assumption. Is there a better way to turn this into a system of first order differential equations implicitly?

$ax=\frac{d}{dt}(f(x,y,t)\frac{dy}{dt}),\phantom{\rule{1em}{0ex}}by=\frac{d}{dt}(g(x,y,t)\frac{dx}{dt})$

For arbitrary functions f and g and constants a and b. I am struggling to find a way to transform this into a system of first order differential equations that I can pass into a solver. It looks like I will need to define these implicitly, but I'm not sure how to do that.

My best attempt so far is the following:

$\begin{array}{rl}{z}_{1}& =f(x,y,t)\frac{dy}{dt}\\ {z}_{2}& =g(x,y,t)\frac{dx}{dt}\\ {z}_{3}& =ax\\ {z}_{4}& =by\end{array}$

${\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right)}^{\prime}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& \frac{a}{g(t,x,y)}& 0& 0\\ \frac{b}{f(t,x,y)}& 0& 0& 0\end{array}\right)\left(\begin{array}{c}{z}_{1}\\ {z}_{2}\\ {z}_{3}\\ {z}_{4}\end{array}\right)$

However, this seems fairly inelegant and assumes that you are always able to divide by f and g. I'm trying to keep this as general as possible, so don't want to make that assumption. Is there a better way to turn this into a system of first order differential equations implicitly?

asked 2022-09-04

How can I solve this differential equation? : $(2{x}^{3}-y)dx+xdy=0$

asked 2022-06-01

Is a first order differential equation categorized by $f({y}^{\prime},y,x)=0$ or ${y}^{\prime}=f(y,x)$?

In the case of the second, why $\mathrm{sin}{y}^{\prime}+3y+x+5=0$ isnt a first order differential equation?

In the case of the second, why $\mathrm{sin}{y}^{\prime}+3y+x+5=0$ isnt a first order differential equation?

asked 2022-06-20

We know that every differential equation is equivalent to a first-order system. I am trying to prove or disprove the converse. For example in ${\mathbb{R}}^{2}$, if we have a system $\dot{x}=f(x,y)$, $\dot{y}=g(x,y)$. Can we always convert it to one differential equation (for example, only in terms of $x$)? Under what condition, this is possible?