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sorry , i mean y and x . How to continue ?Buzz Bloom said:Hi hotjohn:
You may have a typo. You say
V and x are not separablebut the is no "V" in the equation.
Regards,
Buzz
sorry , i didnt get you , can you explain further ?Buzz Bloom said:Hi hotjohn:
If you factor the 2nd equation in your attachment, and make a substitution for y in terms of a new variable, say z, you can get a separable equation involving z and x.
Hope this helps.
Regards,
Buzz
can you expalin why there is a need to sub y = z-1 ?? and how do u knw why should sub y = z-1 ? why can't be y = z-2 ? or others ?Buzz Bloom said:Hi hotjohn:
dy/dx = (1+y) × (1+x2)
y = z-1
Regards,
Buzz
You don't have to sub if you don't want to. Once it's separated just solve it like you would any seperable equation.hotjohn said:can you expalin why there is a need to sub y = z-1 ?? and how do u knw why should sub y = z-1 ? why can't be y = z-2 ? or others ?
how to determine the value of number or new constant to be substituted into the original equation ?Crush1986 said:You don't have to sub if you don't want to. Once it's separated just solve it like you would any seperable equation.
can it be y = z-2 , y = z-3 and etc ??Crush1986 said:Remember, unless you are given initial conditions you will have an infinite amount of answers to most differential equations.
Hi hotjohn:hotjohn said:can you expalin why there is a need to sub y = z-1 ?? and how do u knw why should sub y = z-1 ? why can't be y = z-2 ? or others ?
Buzz Bloom said:Hi hotjohn:
It is not a need, but a convenience.
y=z-1 → z=y+1 → dz/dx =z × (1+x2) →dz/z = (1+x2) dxThis in now the standard form for a separable equation.
Regards,
Buzz
Hi hotjohn:hotjohn said:how do we know that y must be replaced with y=z-1 ?
The answer to this question depends on the specific form of the differential equation. In general, if the equation can be written in the form of dy/dx = f(x)g(y), then it can be separated into two equations for x and y. However, if the equation is not in this form, then it may not be possible to separate V and x.
Separating V and x in a differential equation means rewriting the equation in a form where V and x are on opposite sides of the equation and are not multiplied together. This allows for the solution of the equation to be found by integrating each side separately.
Being able to separate V and x in a differential equation allows for the use of the method of separation of variables, which is a powerful technique for solving certain types of differential equations. It also simplifies the integration process and makes it easier to find a solution to the equation.
The most common technique for separating V and x in a differential equation is the method of separation of variables. Other techniques include using substitution, transforming the equation into a separable form, or multiplying both sides by an integrating factor.
Yes, there are some restrictions and limitations when separating V and x in a differential equation. The equation must be in a certain form (such as dy/dx = f(x)g(y)) in order to be separable. Additionally, there may be cases where separating V and x is not possible or does not lead to a solution that can be easily integrated.