User:Egm6321.f12.team4.lu/Homework2 R2.2&R2.3

Statement
Verify that (2.2.1) is indeed the solution for (2.2.2)

Solution
From (2.2.1), we can derive

Substituting $$\displaystyle p' $$ from (2.2.3) in the left hand side (LHS) of (2.2.2), and we can get

Hence, it is verified.

End of proof.

Author and References

 * Solved and Typed by -- Jinchao Lu

Statement
Show that (2.3.1) is linear in $$ y' $$, and that (2.3.1) is in general an N1-ODE. But (2.3.1) is not the most general N1-ODE as represented by (2.3.2). Give an example of a more general N1-ODE.

i)Show that (2.3.1) is linear in $$ y' $$
Transfer (2.3.1) as represented by $$ M(x,y) $$ and $$ M(x,y) $$

we can have

In addition, we have

Now,

Hence, it is proofed

End of proof

ii)(2.3.1) is in general an N1-ODE
The Eq. (2.3.1) is the 1st order differential equation since the highest order of derivative is 1.

The Eq. (2.3.1) is Ordinary Differential equation because the differential equation includes one dependent variable and its derivatives.

We can define $$ f(x,y) $$

Substitute $$ \alpha y_1 + \beta y_2 $$ in $$ y $$ and derive

Since, in general

Hence

It is proofed.

End of proof

iii)(2.3.1) is not the most general N1-ODE as represented by (2.3.2)
We can see above (2.3.10) and (2.3.11)

If and only if $$ M(x,y) $$ is linear, then (2.3.10) will be

If and only if $$ N(x,y) $$ is equal to constant, then (2.3.11) will be

Hence, (2.3.12) will be

It is proofed.

End of proof

iv)Give an example of a more general N1-ODE
An example of a more general N1-ODE is the following. where

Author and References

 * Solved and Typed by -- Jinchao Lu


 * Loc Vu-Quoc., [[media:Pea1.f12.mtg7.djvu|Mtg 8-2]]