Differential equation x^2*y'=y^2

[Mr. Fest]

Hello,

I have trouble solving the following differential equation.

I am trying to learn how to solve that form of DEs.

The DE is:

x²*dy/dx = y²

There are no initial-value problem, but the solution should be given such that y is defined for all x.

The most important for me is to learn how to solve this type of DE.

I tried solving it using the separable equation way.

x²*dy/dx = y² --> dy/y² = dx/x² and then integrate both sides to get -1/y = -1/x + C --> -y = -x + 1/C --> y = x - 1/C but apparently this doesn't solve the equation...

Hope you guys can help me.

Thanks in advance.


Mr. Fest

 

[hilbert2]

How do you solve the equation $-1/y=-1/x+C$ for $y$? The answer is NOT $-y=-x+1/C$ !

 

[Mr. Fest]

How do you solve the equation $-1/y=-1/x+C$ for $y$? The answer is NOT $-y=-x+1/C$ !

Hi,

You are right. The time was around 2am over here when I wrote that. My brain must have been fried after 10 hours of math work.

The answer is ofc:

-1/y = -1/x + C --> -1 = y(-1/x + C) --> -1/(-1/x + C) = y --> -1/((-1+Cx)/x) --> -x/(-1 + Cx) = y

And, the specific solution, that, y is defined for all real x gives us that C = 0, for if not, then for some x giving Cx = 1, y would not be defined.

So, one could say that the initial-value problem is that y must be defined for all real x giving rise to C = 0 and ultimately y = -x/-1 --> y = x

This satisfies: x^2*dy/dx = y^2 as this means that x^2*1 = x^2 and also, y is defined for all real x.

If there is anything wrong with my answer, please correct me.

 

[vanhees71]

Your implicit solution is correct, i.e., we indeed have
$$\frac{1}{y}=\frac{1}{x}-C=\frac{1-C x}{x}.$$
This gives
$$y(x)=\frac{x}{1-C x}.$$
Now it's easy to check that this indeed satisfies the given differential equation for any value of $C$.

 

[blue_raver22]

us,

Thank you for reaching out and sharing your difficulty with solving this differential equation. Differential equations can be challenging, but with practice and a good understanding of the concepts, they can be solved successfully.

In this case, you are correct in using the separable equation method. However, your integration may have an error. The correct steps to solve this differential equation are as follows:

1. Rewrite the equation in the form of dy/dx = f(x,y):
dy/dx = y^2/x^2

2. Separate the variables:
y^-2 dy = x^-2 dx

3. Integrate both sides:
∫ y^-2 dy = ∫ x^-2 dx

4. Apply the power rule for integration:
-1/y = -1/x + C

5. Rearrange the equation to solve for y:
y = 1/(1/C - x)

6. Simplify:
y = C/(1 - Cx)

This is the general solution to the differential equation. To find the particular solution, you will need to use an initial condition or boundary value. Since you do not have one given in this problem, the solution is in terms of the constant C.

I hope this helps you better understand how to solve this type of differential equation. If you have any further questions, please do not hesitate to ask. Keep practicing and you will become more comfortable with solving differential equations.

Best,