Separable First Order Differential Equations: Solving y'=x√y

In summary, the conversation is about solving a differential equation y'=x√y. The speaker has tried solving it using a certain method and has arrived at the solution y=(x^2+C)/4. They compare this to the answer in their textbook, which is y=±√(x^2+C). They mention that the textbook's answer is for a different differential equation y'=x/y. The conversation also includes a funny exchange about the speaker's name and how they never thought they would discuss ODEs with someone named "dragonblood." The summary concludes by stating that the speaker's solution is correct for the given problem and suggests that the textbook may have mixed up the solutions for different differential equations.
  • #1
dragonblood
22
0
I have tried to solve the differential equation

[tex]y'=x\sqrt{y}[/tex]

like this:

[tex]y^{-\frac{1}{2}}y'=x[/tex]
[tex]\int{y^{-\frac{1}{2}}}dy=\int{xdx}[/tex]
[tex]y^{\frac{1}{2}}=\frac{x^2 +C}{4}[/tex]
[tex]y=\left(\frac{x^2+C}{4}\right)^2[/tex]

Is this the right way to solve it? Because the answer in my textbook says that the answer is
[tex]y=\pm\sqrt{x^2+C}[/tex]

But I really can't see where I've gone wrong.
 
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  • #2
The textbook's answer solves the diff.eq y'=x/y, rather than the one given.

See if your book may have mixed up the ordering of solutions to various diff.eq problems!

(Your own solution is correct for the problem given)
 
  • #3
Yes, I got the same solution as you did, dragonblood.

Wow, never thought I could discuss ODEs with someone named dragonblood.

Cool lol

Matt
 
  • #4
You wouldn't want to know what arildno means in Norwegian, SMLSKDMGLGURU! :smile:
 

Related to Separable First Order Differential Equations: Solving y'=x√y

What is a first order differential equation?

A first order differential equation is an equation that involves an unknown function and its first derivative. It is used to describe relationships between variables and their rates of change.

What does it mean for a first order differential equation to be separable?

A first order differential equation is separable if it can be written in the form of two separate functions, one containing the dependent variable and the other containing the independent variable. This allows for the equation to be solved by integrating both sides separately.

How do you solve a separable first order differential equation?

To solve a separable first order differential equation, first separate the variables on each side of the equation. Then, integrate both sides separately, with respect to the corresponding variable. Finally, solve for the constant of integration and substitute back into the equation to find the general solution.

What is the difference between an explicit and implicit solution to a first order separable differential equation?

An explicit solution is one where the dependent variable is expressed explicitly in terms of the independent variable. An implicit solution is one where the dependent variable is not explicitly expressed, but rather is contained in an implicit function. Both types of solutions are valid and can be used to solve a first order separable differential equation.

Can a first order separable differential equation have more than one solution?

Yes, a first order separable differential equation can have infinitely many solutions. This is because the constant of integration can take on any value, resulting in a different solution for each value. However, a particular initial condition can uniquely determine a specific solution.

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