Edited by Valentin Fadeev, Tuesday, 6 Jul 2010, 23:04
This one gave me some hard time:
where when
The difference with "ordinary" linear equations is that coefficients here depend on . At the same time right hand side is identically zero, so this is not strictly an inhomogeneous equations.
I made false starts trying to find some nice substitute to absorb or alternatively pull out the missing term to construct an inhomogeneous equations. Finally, I found a hint in the book N.M. Günter, Integration of First-Order Partial Differential Equations, ONTI/GTTI, Leningrad/Moscow (1934). In the solution for an equation of a similar structure it was suggested to use the standard method and treat as a constant when integrating the associated system.
So here we go. Searching for a solution in an implicit form:
The associated system:
In the same book it is hinted that one particular integral of this system is which follows from the last identity. Another integral can be found using the first identity:
Treating as constant we can simplify the expressions:
Therefore the general solution can be written in the form:
It is already within reach of sheer guess to let to establish the result, however we proceed with a more lengthy, yet rigorous way.
The system of the first integrals is written as follows:
Using the initial condition :
Solving for and :
Now following the standard method already mentioned below:
MathJax failure: TeX parse error: Extra open brace or missing close brace
Suppressing the solution which does not satisfy intial conditions, finally we obtain:
Gunter's example
This one gave me some hard time:
where when
The difference with "ordinary" linear equations is that coefficients here depend on . At the same time right hand side is identically zero, so this is not strictly an inhomogeneous equations.
I made false starts trying to find some nice substitute to absorb or alternatively pull out the missing term to construct an inhomogeneous equations. Finally, I found a hint in the book N.M. Günter, Integration of First-Order Partial Differential Equations, ONTI/GTTI, Leningrad/Moscow (1934). In the solution for an equation of a similar structure it was suggested to use the standard method and treat as a constant when integrating the associated system.
So here we go. Searching for a solution in an implicit form:
The associated system:
In the same book it is hinted that one particular integral of this system is which follows from the last identity. Another integral can be found using the first identity:
Treating as constant we can simplify the expressions:
Therefore the general solution can be written in the form:
It is already within reach of sheer guess to let to establish the result, however we proceed with a more lengthy, yet rigorous way.
The system of the first integrals is written as follows:
Using the initial condition :
Solving for and :
Now following the standard method already mentioned below:
MathJax failure: TeX parse error: Extra open brace or missing close brace
Suppressing the solution which does not satisfy intial conditions, finally we obtain: