Find the General Solution
Solution:
Solution:
Solution:
A second-order differential equation can be written in the form:
Therefore, the problem given is a second order linear equation.
Here are STEPS on how to get the general solution:
i. simplify the equation to a Second ODE Form
ii. Let
iii. By recalling, we can see that the equation is in First-order linear differential equation form. Solving the simplified equation using FOLDE.
Find the integrating factor
Substituting the I.F. to the formula
Integrating the first term
Let u = 2x and du/2 = dx
By IBP, Let v=u, dv=du and eudu, n=eu.
for the second term
Let u = 2x and du/2 = dx
Combining all the solved terms we get
Based on the equation that we derived it is now a separable differential equation, therefore,
GENERAL SOLUTION:
Solution:
Based on Special Second-Ordered Differential Equation: Special case 3
Denote and substitute to the given equation.
We will have,
Divide both sides with
We will come to,
Tranpose,
We will have
Integrate both sides,
The equation will become a SEPARABLE DIFFERENTIAL EQUATION, multiply both sides with
We will come to the equation:
Integrate both sides,
The answer will be:
Apply logarithmic definition and exponent rule
The answer will be:
Recall that
Substitute the original value of P,
Again, this is a Separable Differential Equation, multiply both sides with:
It will become
Integrate both sides,
The answer will be
Multiply both sides with 3 and the final answer will be
You can still solve it explicitly,
Solutions:
Basically, We need to make the orders of each term to 1. To be able to further break down the equation.
Substituting to the equation, we get
Removing the variables y and P from the 1st term we get
By means of Separation of Variables
We get
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