(a) What can you say about the graph of a solution of the equation y^'=x y^3 when x is close to

(a) What can you say about the graph of a solution of the...

(a) What can you say about the graph of a solution of the equation $y^{\prime}=x y^3$ when $x$ is close to 0 ? What if $x$ is large? (b) Verify that all members of the family $y=\left(c-x^2\right)^{-1 / 2}$ are solutions of the differential equation $y^{\prime}=x y^3$. (c) Graph several members of the family of solutions on a common screen. Do the graphs confirm what you predicted in part (a)? (d) Find a solution of the initial-value problem $$ y^{\prime}=x y^3 \quad y(0)=2 $$

(a) What can you say about the graph of a solution of the equation $y^{\prime}=x y^3$ when $x$ is close to 0 ? What if $x$ is large?
(b) Verify that all members of the family $y=\left(c-x^2\right)^{-1 / 2}$ are solutions of the differential equation $y^{\prime}=x y^3$.
(c) Graph several members of the family of solutions on a common screen. Do the graphs confirm what you predicted in part (a)?
(d) Find a solution of the initial-value problem

$$
y^{\prime}=x y^3 \quad y(0)=2
$$

Key Concepts

Graphical Analysis of Differential Equations

Graphing multiple members of a family of solutions on a common screen allows for visual comparison and validation of the predicted qualitative behavior. Through such plots, one can observe trends like convergence, divergence, or asymptotic behavior across different scenarios.

Initial Value Problems

An initial value problem involves finding a unique solution from a general family of solutions by applying a condition that specifies the value of the unknown function at a particular point. This process of determining the arbitrary constant based on initial conditions is essential for modeling specific real-world situations.

General Solution and Families of Solutions

After integrating a separable differential equation, the result typically includes an arbitrary constant, leading to a family of solutions. This general solution represents all possible behaviors of the system depending on the values of the constant, and it is foundational in understanding the solution space.

Qualitative Analysis of Differential Equations

This involves understanding how the solutions behave without necessarily solving the equation completely. For instance, analyzing the dynamics near specific values (such as when the independent variable is small) or as it becomes large helps in predicting the shape, slopes, and trends of the solution curves.

Separable Differential Equations

A separable differential equation is one in which the variables can be rearranged so that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side. This property allows the equation to be integrated directly and leads to a family of solutions parameterized by an arbitrary constant.

Verification of Proposed Solutions

Verifying a proposed solution means substituting it back into the differential equation to check if it satisfies the equation for all values in its domain. This step is crucial for confirming that the derived family of functions indeed forms valid solutions.

Transcript

00:01 In this video, we're going to be looking at solution forms for differential equations.

00:04 So the equation we're looking at is y prime equals x times y cubed.

00:09 So let's first think about what a graph of the solution to this equation would look like for small x values.

00:15 So looking at the right -hand side of this equation, we can see that a small x value will correlate to a small slope.

00:29 Or in other words, as x goes to zero, y prime goes to zero.

00:40 And similarly for large x values, a large x value will correlate to a large slope.

00:55 Or as x goes to infinity, our y prime will also go to infinity because y prime and x prime and x, are directly proportional.

01:12 So now, if we're given the solution form of y equals the quantity c minus x squared to the negative 1 half, how can we go about verifying that that is a solution to this differential equation here? so we know why we can directly plug that in.

01:33 What we need to do now is find y prime so that we can plug that in and see whether or not this solution form will work.

01:43 So let's find y prime by taking the derivative of y.

01:47 So the derivative of this will be the power that this quantity is raised to, negative one half, times the quantity to a power that is one less than the power it has already raised to.

02:00 So that's going to be negative 3 over 2 times the derivative of the quantity or negative 2x.

02:09 So if we go ahead and simplify that, our negative signs cancel, our 2s cancel, and we get y prime equals x times c minus x squared to the negative 3 over 2.

02:25 So let's plug in this and our y into the differential equation and see if we get a true statement.

02:31 So x equals, oh, whoops, x times c minus x squared to the negative three halves equals x times y, c minus x squared to the negative one half to the third.

02:56 And we know that power raise to power just means that we take these two and multiply.

03:06 So let's go ahead and do that.

03:08 Simplify, we get x times c minus x squared to the negative three halves equals x times c minus x squared to the negative three halves.

03:21 So this is a true statement and therefore this solution form is valid...

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