- #1
karush
Gold Member
MHB
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$ y'= \dfrac{x^2}{y(1+x^3)}$
Separate y dy =\dfrac{x^2}{(1+x^3)...
ok i tried to get the book ans but someahere derailed
why is =c in the answer
Huh? What step are you not understanding?karush said:ok i don't see how you can just
put dx in separations
The concept of separating variables in solving differential equations involves isolating the dependent and independent variables on opposite sides of the equation to make it easier to solve. This is done by rearranging the equation and integrating both sides separately.
The method of separating variables is typically used when the differential equation is in the form of dy/dx = f(x)g(y), where f(x) and g(y) are functions of the independent and dependent variables, respectively. This method can also be used when the equation can be manipulated into this form.
The steps involved in solving a differential equation by separating variables are as follows:
1. Rearrange the equation so that all terms with the dependent variable are on one side and all terms with the independent variable are on the other side.
2. Integrate both sides of the equation separately.
3. Add a constant of integration to one side of the equation.
4. Solve for the dependent variable to get the general solution.
5. If initial conditions are given, use them to find the particular solution.
Some common mistakes to avoid when using the method of separating variables include:
- Forgetting to add a constant of integration
- Incorrectly integrating one or both sides of the equation
- Not isolating the dependent and independent variables on opposite sides of the equation
- Forgetting to substitute the initial conditions into the particular solution
- Making algebraic errors during the solving process.
No, the method of separating variables can only be applied to certain types of differential equations, specifically those in the form of dy/dx = f(x)g(y). Other types of differential equations may require different methods, such as substitution or using an integrating factor.