U-Substitution, can't figure out where the answer comes from?

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In summary, the question asks to solve the differential equation dy/dx = y^2-2y+1 using U-Substitution. The answer key provides the solution y = 1+(x+C)^-1. The student attempted to solve it by rearranging the equation and ended up with x = ln(y^2)^2y-ln(2y)^2+y. However, the correct solution involves factoring y^2-2y+1. After receiving a hint, the student was able to solve the problem.
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shft600
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Homework Statement


Simply enough, solve the following using U-Substitution.

dy/dx = y2-2y+1

and the answer key provides

y = 1+(x+C)-12. The attempt at a solution

I know how to do this, I thought. But I'm just spinning my wheels and need somebody to show me my error...

I tried moving around the dy/dx parts, so that I eventually ended up with
dx=dy/(y2-2y+1)
which simplified to
x=ln(y2)2y-ln(2y)2+y, but that's not right, unless I'm missing some kind of simplification?

Honestly, I have no idea where the textbook got the answer. Any tips? What am I doing wrong?
 
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  • #2
Can you factor y^2-2y+1?
 
  • #3
That would be it! I knew it was something simple, thanks!
 

1. What is U-Substitution?

U-Substitution is a technique used in calculus to evaluate integrals by replacing a complicated expression with a simpler one.

2. How does U-Substitution work?

To use U-Substitution, you first identify a portion of the integrand (the expression being integrated) that looks similar to the derivative of another function, which we call U. You then substitute U into the integral and also replace any other occurrences of the original function with U. This allows you to simplify the integral and make it easier to evaluate.

3. Why is U-Substitution useful?

U-Substitution is useful because it allows us to evaluate integrals that would otherwise be difficult or impossible to solve. It also helps us to understand the relationship between derivatives and integrals.

4. What is the difference between U-Substitution and integration by parts?

U-Substitution and integration by parts are both techniques used to evaluate integrals, but they are used in different situations. U-Substitution is used when the integrand contains a single variable that can be replaced with a simpler expression, while integration by parts is used when the integrand is a product of two functions. U-Substitution is often used for integrals involving trigonometric functions, while integration by parts is useful for integrals involving polynomial or exponential functions.

5. Can you provide an example of U-Substitution in action?

Sure, let's look at the integral ∫(x^2 + 2x)^3 dx. We can use U-Substitution by letting U = x^2 + 2x. This means that dU/dx = 2x + 2, or 2x = dU/dx - 2. Substituting this into the integral, we get ∫(U)^3 (dU/dx - 2)dx. Simplifying and integrating, we get (1/4)U^4 - (1/2)U^3 + C. Substituting back in x for U, we get the final answer of (1/4)(x^2 + 2x)^4 - (1/2)(x^2 + 2x)^3 + C.

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