Question

How do you find all solutions of the differential equation $\dfrac{{{d}^{3}}y}{d{{x}^{3}}}={{e}^{x}}?$

Answer 1

As we have the given differential equation. It is the third order of separable differential equation in which we can solve it by using the integration. Or simply we can separate the variable. Now we have to integrate the given differential equation three times due to this we get the three equations. At last we have to combine all of the equations and write it which is the answer to our questions.

Complete step by step solution:
The given differential equation.
$\dfrac{{{d}^{3}}y}{d{{x}^{3}}}={{e}^{3}}$
We can see that this is the third order separable differential equation.
In this type of differential equation we can solve it by integrating the given equation three times (or also we say it by separating the variables).
$\therefore$ By integrating given equation first time we get, $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}\,\,=\,\,{{e}^{x}}\,\,+\,A\,\,.............\left( i \right)$
Now, we have to integrate it by second time we get, $\dfrac{dy}{dx}\,\,=\,{{e}^{x}}\,\,+\,{{A}_{1}}\,x\,\,+B\,\,\,..........\left( ii \right)$
Third time we integrate the above equation we get, $y\,=\,{{e}^{x}}+\,{{A}_{1}}\,\,\dfrac{{{x}^{2}}}{2}\,+\,Bx\,\,+C\,\,\,.............\left( iii \right)$
So by combining all the three equation and write the GS as:
$y\,=\,{{e}^{x}}\,+\,A{{x}^{2}}\,\,+\,Bx\,+C$.
This is the required solution for the given differential equation.

Note: First of all understand the part of the question that is asking. Then read the question carefully and solve that first in which it is difficult for you to solve. As you are reading the answer try to think of the answer before you see any options. If you are expert in writing lengthy answers such as short answers this strategy will be of greater help for us. In the option try to eliminate the wrong option. Also underline the key part of the question. Don’t read the question too much. These are the points you have to remember while solving any questions.