# First Order Non-Linear Differential Equation

• Oct 3rd 2010, 10:27 AM
Belowzero78
First Order Non-Linear Differential Equation
Question: Solve the Initial value problem:

$\displaystyle (5t^4y^{-3}+7t^6y^7)dt + (-3t^5y^{-4} + 7t^7y^6)dy = 0, y(1) = 1$.

Express your answer in the form $\displaystyle F(t,y) = 2$, where $\displaystyle F(t,y)$ has no constant term.

$\displaystyle F(t,y)=$ __________________ $\displaystyle = 2$

Attempt at the question:

First of all I checked the terms to see if they were exact and they are not. I took the integral with respect to $\displaystyle t$ for $\displaystyle M(t,y)$ which is the first part of the equation and it came out to be $\displaystyle t^5y^{-3} + t^7y^7 + k(y)$. Afterwards I took the partial with respect to y of that integral we just got which was: $\displaystyle -3t^5y^{-4} + 7t^6y^7 + k'(y)$. I now would equate both these equations together and would cancel terms. The result gives me $\displaystyle k'(y) = 0$. So, my equation $\displaystyle F(t,y) = t^5y^{-3} + t^7y^7$. But my question now is, how do I solve using the initial values since my equation has both terms $\displaystyle t$ and $\displaystyle y$?

Thanks for everyone who contributes!
• Oct 3rd 2010, 10:44 AM
TheEmptySet
This is an implicitly defined curve. You are told that the solution should look like

$\displaystyle F(y,t)=2$ this must hold for all y and t

You have $\displaystyle F(y,t)=\frac{t^5}{y^3}+t^7y^7=2=F(1,1)$ so the implict solution is

$\displaystyle \frac{t^5}{y^3}+t^7y^7=2$ we can check this by taking the derivative

$\displaystyle 5t^4y^{-3}-(3t^5y^{-4})\frac{dy}{dt}+7t^6y^7+(7t^7y^6)\frac{dy}{dt}=0$

$\displaystyle (-3t^5y^{-4}+7t^7y^6)dy+(5t^4y^{-3}+7t^6y^7)dt=0$
• Oct 3rd 2010, 10:46 AM
Belowzero78
Ok, so my answer that i reached is correct. Essentially, what does no constant term mean? I just want to clarify that for my understanding!