Find the general solution for each differential equation. Verify that each solution satisfies th

step 1 I have D .Y over DX equals E to the Y squared all over Y. This is a differential equation becaus

Find the general solution for each differential equation....

Key Concepts

Verification of Solutions

After obtaining the general solution through integration, it is critical to verify that the solution satisfies the original differential equation. This typically involves differentiating the general solution with respect to the independent variable and checking that the result matches the given differential equation. This step ensures the mathematical validity of the derived solution.

General Solutions and the Integration Constant

The process of integrating both sides of a separable differential equation introduces an arbitrary constant, which represents the family of all possible solutions to the differential equation. In this case, after integrating the separated variables, the general solution includes an integration constant that accounts for the initial conditions and the generality of the solution.

Separable Differential Equations

This concept involves recognizing when a differential equation can be written in the form f(y) dy = g(x) dx, allowing the variables to be separated onto different sides of the equation. In this context, the given differential equation is manipulated to isolate terms involving y on one side and x on the other, which then permits independent integration of both sides.

Integration Techniques (Substitution)

When the separated equation involves nontrivial integrals, substitution methods are often used to simplify the integration process. In the given differential equation, the exponent y^2 suggests a substitution (for example, letting u = y^2) may ease the integration, illustrating how substitution is a vital tool for integrating functions that are compositions of simpler functions.

Transcript

00:01 I have d .y over dx equals e to the y squared all over y.

00:05 This is a differential equation because i have that dy over dx.

00:08 And if i want to get it back to a general solution, then i need to do the integral.

00:12 But first i would want to separate the variables.

00:15 So i would send the dx over to the right side because that gets it off the denominator.

00:20 But i noticed that really i have the wrong variable there with it.

00:25 So i'm going to need to move this entire rational over to the left side.

00:30 An easy way to do that is to use the reciprocal.

00:33 So if i use the reciprocal, that's me flipping that fraction so that this will cancel out.

00:40 So that reciprocal to keep things balanced goes over to the left side.

00:46 So my new line of work is going to be y over e to the y squared times dy equals just dx or i could write 1dx.

00:58 Now that my variables are separated, i'm ready to integrate.

01:04 Integrating on the left hand side, i want to do one important step first.

01:11 There's not like a great rule for having an e on the denominator.

01:16 So i actually can bring that e up to the numerator if i write the exponent as a negative, right? that will reciprocated for me.

01:26 So i'm going to write that exponent as a negative y squared to bring it up and everything else is going to stay the same.

01:30 And now it becomes more obvious to me that i can use a substitution here to make it just e to the u, which its anti -derivative stays the same.

01:40 It's still e to the u.

01:41 So we're going to replace this exponent, negative y squared, and i need to also check its derivative.

01:50 Well, the derivative would be negative 2y, d .y.

01:56 And i have some of that already here...

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