# Little help with differential equations

• Titans86
In summary, the conversation discusses solving a differential equation using integration and the application of an initial condition. The solution involves finding a general solution and using the initial condition to find a particular solution. There may be multiple particular solutions and it is important to verify that the solution satisfies the differential equation and initial condition.
Titans86

## Homework Statement

$$\frac{dy}{dx} = \sqrt{xy^3} , y(0) = 4$$

## The Attempt at a Solution

So;

$$\frac{dy}{dx} = x^{\frac{1}{2}}y^{\frac{3}{2}} \Rightarrow \int y^{-\frac{3}{2}}dy = \int x^{\frac{1}{2}}dx \Rightarrow -2y^{-\frac{1}{2}} = \frac{2}{3}x^{\frac{3}{2}} + C \Rightarrow y^{-\frac{1}{2}} = -\left(\frac{1}{3}x^{\frac{3}{2}} + \frac{1}{2}C\right) \Rightarrow y^{\frac{1}{2}} = -\left(\frac{1}{\frac{1}{3}x^{\frac{3}{2}} + \frac{1}{2}C}\right) \Rightarrow y = \frac{1}{(\frac{1}{3}x^{\frac{3}{2}} + \frac{1}{2}C)^2}$$

Then Putting in the conditions mentioned above:

$$4 = \frac{1}{(0 + \frac{1}{2}C)^2} \Rightarrow \frac{1}{4}C^2 = \frac{1}{4} \Rightarrow C = 1$$

Yet my book shows $$C = \frac{3}{2}$$

Your work looks fine to me except near the end. C can also equal -1.
Is the book's answer y = 1/[1/3*x^(3/2) + 1/2 * 3/4]^2? (I.e., same as yours but with C = 3/2 rather than C = 1 as you have it.)

I don't see why your general solution wouldn't be a solution to the DE, and I don't see any problems with your use of the initial condition to find the particular solution. I would want to verify that the book's solution solves the DE and initial condition. After all, math text occasionally have typos.

you mean there can be more then one particular solution?

## 1. What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes over time, based on the rate at which the quantity is changing. They involve derivatives, which represent the rate of change of a function.

## 2. Why are differential equations important?

Differential equations are used to model and understand many natural phenomena, such as population growth, disease spread, and chemical reactions. They are also essential in many fields of science, including physics, engineering, and economics.

## 3. How do I solve a differential equation?

There are various techniques for solving differential equations, depending on the type of equation. Some common methods include separation of variables, substitution, and using an integrating factor. It is also possible to use numerical methods, such as Euler's method, to approximate solutions.

## 4. Can differential equations have multiple solutions?

Yes, differential equations can have multiple solutions. This is known as the "general solution." However, for many real-world problems, we are interested in finding a specific solution that satisfies certain initial conditions.

## 5. What are the applications of differential equations?

Differential equations are used in a wide range of applications, including physics, engineering, biology, economics, and even social sciences. They are particularly useful in modeling and predicting the behavior of systems that change over time.

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