Separable Differential Equation with Initial Value?
#x^2(dy/dx)=yxy#
Initial condition : y(1)=1
This is what I separated it to
#(dy/y)=(1x)/(x^(2))dx#
After integration, I have:
#ln(y)=(xln(x)1)/(x)#
I know that I will have to introduce "e," but I am getting tripped up here.
Any help would be appreciated.
Thanks!
Initial condition : y(1)=1
This is what I separated it to
After integration, I have:
I know that I will have to introduce "e," but I am getting tripped up here.
Any help would be appreciated.
Thanks!
# y = (e^(1/x1))/(x) #
We have:
which we could type as:
Since the ODE can be divided, as shown, we can "separate the variables" to obtain:
Since both integrals pertain to standard functions, we can integrate them directly to obtain:
Consequently, we have:
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A separable differential equation is one that can be expressed in the form ( \frac{dy}{dx} = f(x)g(y) ), where ( f(x) ) is a function of ( x ) and ( g(y) ) is a function of ( y ). To solve a separable differential equation with an initial value, you typically follow these steps:

Separate the variables: Rewrite the equation so that all terms involving ( y ) are on one side and all terms involving ( x ) are on the other side.

Integrate both sides: Integrate both sides of the equation with respect to their respective variables.

Solve for ( y ): After integrating, you'll have an equation involving ( y ) and ( x ). Solve this equation for ( y ) to find the general solution.

Apply the initial condition: Use the initial value given to find the particular solution by substituting the initial value into the general solution and solving for any constants.

Check for any restrictions: Ensure that the solution obtained is valid for all values of ( x ) within the domain of the original problem.
This process yields the solution to the separable differential equation with the given initial value.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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