Recent questions in First order differential equations

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Hailee S.2022-12-19

The rate at which a body cools is proportional to the difference in

temperature between the body and its surroundings. If a body in air

at 0℃ will cool from 200℃ 𝑡𝑜 100℃ in 40 minutes, how many more

minutes will it take the body to cool from 100℃ 𝑡𝑜 50℃ ?

Differential EquationsOpen question

Hailee S.2022-12-19

A body falls from rest against a resistance proportional to the velocity at any instant. If the limiting velocity is 60fps and the body attains half that velocity in 1 second, find the initial velocity.

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vegetatzz8s 2022-11-25

How can I solve this differential equation? : $xydx-({x}^{2}+1)dy=0$

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Ty Moore 2022-11-18

What is the general solution of the differential equation? : ${x}^{2}\frac{{d}^{2}y}{{dx}^{2}}+x\frac{dy}{dx}+{n}^{2}y=0$

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Uroskopieulm 2022-11-16

Solve the Differential Equation $\frac{dy}{dx}=2xy-x$?

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Aron Heath 2022-11-05

What is a solution to the differential equation $xy\prime +2y=0$?

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Trace Glass 2022-10-27

${x}^{2}{y}^{\u2033}+x{y}^{\prime}-y={x}^{2}$

My attempt:

Divided by ${x}^{2}$:

${y}^{\u2033}+\frac{{y}^{\prime}}{x}-\frac{y}{{x}^{2}}=1$

Now to solve the homogenous equation using Euler's method

${y}^{\u2033}+\frac{{y}^{\prime}}{x}-\frac{y}{{x}^{2}}=0$

To look for solution $y={x}^{r}$

so ${y}^{\prime}=r{x}^{r-1}$

${y}^{\u2033}=r(r-1){x}^{r-2}$

So:

$r(r-1){x}^{r-2}+\frac{r{x}^{r-1}}{x}-\frac{{x}^{r}}{{x}^{2}}=0$

Divided by ${x}^{r}$:

$r(r-1){x}^{-2}+\frac{r{x}^{-1}}{x}-\frac{1}{{x}^{2}}=0$

Is it correct so far?

My problem: I don't know how to find ${r}_{1},{r}_{2}$

My attempt:

Divided by ${x}^{2}$:

${y}^{\u2033}+\frac{{y}^{\prime}}{x}-\frac{y}{{x}^{2}}=1$

Now to solve the homogenous equation using Euler's method

${y}^{\u2033}+\frac{{y}^{\prime}}{x}-\frac{y}{{x}^{2}}=0$

To look for solution $y={x}^{r}$

so ${y}^{\prime}=r{x}^{r-1}$

${y}^{\u2033}=r(r-1){x}^{r-2}$

So:

$r(r-1){x}^{r-2}+\frac{r{x}^{r-1}}{x}-\frac{{x}^{r}}{{x}^{2}}=0$

Divided by ${x}^{r}$:

$r(r-1){x}^{-2}+\frac{r{x}^{-1}}{x}-\frac{1}{{x}^{2}}=0$

Is it correct so far?

My problem: I don't know how to find ${r}_{1},{r}_{2}$

Differential EquationsOpen question

Jane Galura2022-10-26

A metal is heated-up to a temperature of 500 degree Celsius. It is then exposed to a temperature of 38 degree Celsius. After 2 minutes, the temperature of the metal becomes 190 degree Celsius.

a. when will the temperature be 100°C?

b. what is the temperature after 4 minutes?

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benatudq 2022-10-22

What is a second solution to the Differential Equation ${x}^{2}y\prime \prime -3xy\prime +5y=0$?

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erwachsenc6 2022-10-12

How do you solve the differential equation given $f\prime \left(s\right)=6s-8{s}^{3}$?

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blackdivcp 2022-10-11

What is a solution to the differential equation $y\prime =2x+1$?

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smetuwh 2022-09-30

How to you find the general solution of $\frac{dy}{dx}=x{e}^{{x}^{2}}$?

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Ratuiszt 2022-09-30

How do I solve the differential equation $xy\prime -y=3xy,{y}_{1}=0$?

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Parker Pitts 2022-09-30

What is a solution to the differential equation $\frac{dy}{dx}=\frac{6\mathrm{sin}x}{y}$?

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Jamarcus Lindsey 2022-09-30

How to do the ordinary differential equation? $\frac{dA}{dt}=\stackrel{.}{A}=-\frac{\alpha}{2m}A\to A\left(t\right)=A\left(0\right){e}^{-\frac{\alpha}{2m}t}$

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mocatgesex 2022-09-30

What is the solution of the differential equation? : $y\frac{dy}{dx}={(9-4{y}^{2})}^{\frac{1}{2}}$ where x=0 when y=0

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timberwuf8r 2022-09-30

How to you find the general solution of $\frac{dr}{ds}=0.05r$?

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jhenezhubby01ff 2022-09-30

What is the general solution of the differential equation $xyy\prime ={x}^{2}+1$ ?

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eukrasicx 2022-09-30

How do you solve $\frac{dy}{dx}=x{y}^{2}$?

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eukrasicx 2022-09-30

What is a general solution to the differential equation $\frac{dy}{dx}=\frac{{x}^{3}}{{y}^{2}}$?

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