MHB Owen b's question at Yahoo Answers regarding a first order homogenous ODE

  • Thread starter Thread starter MarkFL
  • Start date Start date
  • Tags Tags
    First order Ode
AI Thread Summary
The discussion focuses on solving the first-order homogeneous ordinary differential equation (ODE) given by dy/dt = t^3/y^3 + y/t. The approach involves recognizing it as a Bernoulli equation and applying the substitution u = y/t, which simplifies the equation. After substituting, the ODE is transformed into a separable form, allowing for integration. The integration leads to an implicit solution expressed as y^4 = t^4(ln(t^4) + C). This method effectively demonstrates the process of solving the given ODE.
MarkFL
Gold Member
MHB
Messages
13,284
Reaction score
12
Here is the question:

How to solve this equation? dy/dt= t^3/y^3 + y/t?


How to solve this equation? dy/dt= t^3/y^3 + y/t

what i understand is we have to use Bernoulli and then solve it using linear equation,is it?

I have posted a link there to this thread so the OP can view my work.
 
Mathematics news on Phys.org
Hello owen b,

We are given to solve:

$$\frac{dy}{dt}=\frac{t^3}{y^3}+\frac{y}{t}$$

I would first express the ODE as:

$$\frac{dy}{dt}=\left(\frac{y}{t} \right)^{-3}+\frac{y}{t}$$

Now, use the substitution:

$$u=\frac{y}{t}\implies y=ut\implies\frac{dy}{dt}=u+\frac{du}{dt}t$$

And our ODE become:

$$u+\frac{du}{dt}t=u^{-3}+u$$

$$\frac{du}{dt}t=u^{-3}$$

Separating variables and integrating (noting $t\ne0$), we obtain:

$$\int u^3\,du=\int\frac{dt}{t}$$

$$\frac{u^4}{4}=\ln|t|+C$$

$$u^4=\ln\left(t^4 \right)+C$$

Back-substituting for $u$, we obtain the implicit solution:

$$y^4=t^4\left(\ln\left(t^4 \right)+C \right)$$
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top