How do you find all solutions of the differential equation #(d^3y)/(dx^3)=e^x#?

Question

Emilia Aguilar · Accepted Answer

# 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): # (d^3y)/(dx^3)=e^x # Integrating we get # (d^2y)/(dx^2)=e^x + A# And a second time: # (dy)/(dx) = e^x + A_1x +B# And a third time: # y = e^x + A_1x^2/2 +Bx+ C# So we can write the GS as; # y = e^x + Ax^2 +Bx+ C#

Ezekiel Alexander · Answer

undefined To find all solutions of the differential equation $ \frac{d^3y}{dx^3} = e^x $, you can follow these steps:

1. Integrate $ e^x $ three times with respect to $ x $ to find the general solution.

2. After each integration, you'll obtain a constant of integration. You'll end up with a third-degree polynomial equation.

3. Solve the polynomial equation for the constants of integration.

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

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

6. 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.