Did I solve this diff eq substition problem properly?

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In summary, there are a few ways to check if you have solved a differential equation substitution problem properly. The first step in solving a differential equation with substitution is to identify the dependent and independent variables in the equation. Typically, you want to choose the variable that has a derivative given in the equation. Substitution is only applicable for certain types of differential equations, specifically separable equations. To improve your problem-solving skills for differential equations with substitution, it is recommended to practice solving a variety of problems and review the steps and strategies for solving them.
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utnip123
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[itex]xy'=yln(xy)[/itex]
[itex]xdy=yln(xy)dx[/itex]
[itex]\frac{dy}{y}[/itex]=[itex]\frac{ln(xy)dx}{x}[/itex]

Substitution:

[itex]v=ln(xy)[/itex]
[itex]dv[/itex] = [itex]\frac{dy}{y}[/itex]-[itex]\frac{dx}{x}[/itex]
[itex]dv[/itex]-[itex]\frac{dx}{x}[/itex]=[itex]\frac{vdx}{x}[/itex]

∫[itex]\frac{dv}{v+1}[/itex]=∫[itex]\frac{dx}{x}[/itex]

[itex]ln(v+1)=ln x + C[/itex]

v+1 = Cx
ln(xy) +1 = Cx

Would that basically be the complete answer?
 
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  • #2
It might be interesting to solve this last equation for y=... and verify the answer with the initial equation.
 

1. How do I know if I solved a differential equation substitution problem correctly?

There are a few ways to check if you have solved a differential equation substitution problem properly. First, you can plug your solution back into the original equation to see if it satisfies the equation. Additionally, you can use a graphing calculator or software to plot both the original equation and your solution and see if they intersect at the same points. Finally, you can check your answer with a solution manual or ask a peer or instructor for confirmation.

2. What is the first step in solving a differential equation with substitution?

The first step in solving a differential equation with substitution is to identify the dependent and independent variables in the equation. This will help determine which variable to substitute and solve for in order to separate the variables and solve the equation.

3. How do I choose which variable to substitute in a differential equation substitution problem?

Typically, you want to choose the variable that has a derivative given in the equation. This will make the substitution process easier and more straightforward. If both variables have derivatives, you can choose either one and see which substitution leads to a simpler equation to solve.

4. Can I use substitution to solve any type of differential equation?

No, substitution is only applicable for certain types of differential equations, specifically separable equations. These are equations where it is possible to separate the variables and solve for each individually. Substitution may also be used for linear and homogeneous equations, but the process may be more complicated.

5. How can I improve my problem-solving skills for differential equations with substitution?

The best way to improve your problem-solving skills for differential equations with substitution is to practice solving a variety of problems. Start with simpler equations and work your way up to more complex ones. Additionally, it can be helpful to review the steps and strategies for solving these types of equations, such as identifying the variables, making the substitution, and solving for the constant of integration.

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