How do you find all solutions of the differential equation #(d^3y)/(dx^3)=e^x#?
# y = e^x + Ax^2 +Bx+ C#
This is a third order separable differential equation which we can solver by repeated integration, (or separating the variables):
Integrating we get
And a second time:
And a third time:
So we can write the GS as;
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To find all solutions of the differential equation ( \frac{d^3y}{dx^3} = e^x ), you can follow these steps:

Integrate ( e^x ) three times with respect to ( x ) to find the general solution.

After each integration, you'll obtain a constant of integration. You'll end up with a thirddegree polynomial equation.

Solve the polynomial equation for the constants of integration.

Once you have the constants, combine them with the integrated terms to form the general solution of the differential equation.

The general solution will contain all possible solutions to the given differential equation.

Make sure to check the solution for correctness by differentiating it three times with respect to ( x ) to ensure it satisfies the original differential equation.
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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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