System of partial DE's

[menco]

1. The problem statement, all variables and given/known data

Solve the following system of partial differential equations for u(x,y)

2. Relevant equations

\(\displaystyle du/dy = 2xyu \)

\(\displaystyle du/dx = (y^2 + 5)u \)

3. The attempt at a solution

I am honestly not sure where to start, my lectures and tutorials this week have not been helpful at all. My guess is to take the derivative of the first equation and sub that into the second equation for y and then take the derivative of the second equation to get my final answer. But I am probably completely wrong. Any help or advice would be appreciated!

 

[DrPhil]

1. The problem statement, all variables and given/known data

Solve the following system of partial differential equations for u(x,y)

2. Relevant equations

\(\displaystyle \partial u/\partial y = 2xyu \)

\(\displaystyle \partial u/\partial x = (y^2 + 5)u \)

3. The attempt at a solution

I am honestly not sure where to start, my lectures and tutorials this week have not been helpful at all. My guess is to take the derivative of the first equation and sub that into the second equation for y and then take the derivative of the second equation to get my final answer. But I am probably completely wrong. Any help or advice would be appreciated!

Perhaps you meant to suggest integrating the first equation, rather than differentiating?

Separating variables,
\(\displaystyle \dfrac{\partial u}{u} = 2xy \partial y \)
which integrates to
\(\displaystyle \ln{|u|} = xy^2 + f(x) \)
where the "constant of integration" for the partial with respect to y is any function f(x).

Do the same procedure on the partial with respect to x. Does that give
\(\displaystyle \ln{|u|} = xy^2 + 5x + g(y) \) ?
Does the identity of these two equations prove that f(x) = 5x+C1,
and g(y)=C2 ?

Is thee final result
\(\displaystyle \ln{|u|} = xy^2 + 5x + C \) ?
\(\displaystyle u(x,y) = A\ \mathrm e^{xy^2} \mathrm e^{5x} \) ?

 

[HallsofIvy]

1. The problem statement, all variables and given/known data

Solve the following system of partial differential equations for u(x,y)

2. Relevant equations

\(\displaystyle du/dy = 2xyu \)

So, treating x as a constant, du/u= 2xy dy which gives \(\displaystyle ln(u)= xy^2+ f(x)\).
(Since we are treating x as a constant the "constant of integration" may be any function of x- that's f(x).)

\(\displaystyle du/dx = (y^2 + 5)u \)

Differentiating the above formula for ln(u) with respect to x gives \(\displaystyle \frac{1}{u}\frac{\partial u}{\partial x}= y^2+ f'(x)\) and
comparing to this equation, we must have f'(x)= 5 so that f(x)= 5x+ C. That means that \(\displaystyle ln(u)= xy^2+ 5x+ C\). If you want to solve for x, take the exponential of both sides: \(\displaystyle u(x,y)= e^{xy^2+ 5x+ C}= C'e^{xy^2+ 5x}\) where \(\displaystyle C'= e^C\).

3. The attempt at a solution

I am honestly not sure where to start, my lectures and tutorials this week have not been helpful at all. My guess is to take the derivative of the first equation and sub that into the second equation for y and then take the derivative of the second equation to get my final answer. But I am probably completely wrong. Any help or advice would be appreciated!