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

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

Key Concepts

General Solution and Constant of Integration

The general solution of a differential equation includes all possible particular solutions and is expressed in terms of an arbitrary constant (or constants). This constant represents the infinite number of solutions that vary according to initial conditions or boundary values, and it plays an essential role in capturing the overall behavior dictated by the differential equation.

Verification of Solutions

Verifying a solution of a differential equation involves substituting the obtained general solution back into the original equation to confirm that it satisfies the relationship for all values of the independent variable. This practice is important to ensure the correctness of the solution and to validate the integration steps and manipulations performed during the solving process.

Ordinary Differential Equations

An ordinary differential equation (ODE) is an equation that involves functions of a single variable and their derivatives. In solving ODEs, the goal is often to find the general solution, which represents a family of functions that satisfy the equation for all values in the domain, typically including an arbitrary constant.

Integration and Antiderivatives

Integration is the process of finding an antiderivative, which is the inverse operation of differentiation. In the context of solving differential equations, integration is used to recover the original function from its derivative, with an inclusion of an arbitrary constant to represent the family of all possible solutions.

Transcript

00:01 So let's say we're looking at this differential equation, 3x squared minus 3, d, y, over dx equals 2.

00:06 And if i want to take it back to the general solution, then i want to get rid of that dy over dx.

00:12 So actually, my first move would be to try and isolate dy over dx.

00:17 So i would subtract that 3x squared to the other side, and then i would divide by the negative 3.

00:27 So dy over dx is by itself, dividing by the negative 3 gives me a 1.

00:32 X squared minus two thirds, dividing both terms by negative three.

00:37 So at this point, i am ready to separate the variables.

00:42 To do that, i'm going to multiply the dx to the other side.

00:46 That allows me to continue to reverse the process of the derivative here by taking the anti -derivative.

00:55 So on the left, i have the integral of d -y, which is like one times d -y.

01:00 So that integral goes to y equals.

01:02 And then on the right side, the anti -derivative here, i would add one to the exponent and then put that exponent, that new exponent on the denominator.

01:14 I'm doing the reverse of the power rule for the antiderivative.

01:17 So one -third x cubed.

01:19 And then minus two -thirds x.

01:21 Remember if you have just a constant in the integral, it becomes the coefficient for your anti -derivative...

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