solving differential equations
Please see the attached file for full description.
1. Find both the first and second order differentials (y' and y") for the following functions:
2. Use integrating factor to convert the following equation into "exact ODE" form and solve for y.
2xy' = (y -x) (y + x)/ y
3. Solve the differential equation, y is a function of x
y'' - 6y' + 9y = x^2x
4. Solve the differential equation, y is a function of x
2y'' - 2y' + 5y = cos(x), y(0) = 1, y'(0) = 2
5. Solve the differential equation, y is a function of x
y'= 2x + 2xy, y(1)= 2
https://brainmass.com/math/calculus-and-analysis/solving-differential-equations-433751
SOLUTION This solution is FREE courtesy of BrainMass!
ET 7430
1. Find both the first and second order differentials (y' and y") for the following functions:
ï€ ï€ ï€ ï€ ï€
a.
Using the product rule and chain rule:
Then the second derivative is:
b.
Then the second derivative is:
c.
Apply the chain rule:
Apply the product rule and chain rule:
2. Use integrating factor to convert the following equation into "exact ODE" form and solve for y.
Rearrange the equation:

Then
And
is a function of x alone, then
is an integrating factor.
So multiplying 1/x2 to the equation (1):
(2)
Whose solution is obtained as follows:
Then
Where is the coefficient of dx in equation (2)
So
Therefore,
And
Which implies that
And the general solution is
3. Solve the differential equation, y is a function of x
The characteristic equation of the homogeneous ODE is
It has double roots
Then the general solution to the homogeneous ODE is
Now, since the right side function is an exponential and we know that exponentials never just appear or disappear in the differentiation process it seems that a likely form of the particular solution would be
Now, all that we need to do is do a couple of derivatives, plug this into the differential equation and see if we can determine what A needs to be.
Then
So a particular solution to the differential equation is then,
Thus, the general solution to the differential equation is
4. Solve the differential equation, y is a function of x
For this undetermined coefficient type of ODE, since the right-hand function is , we guess that the particular solution is
Now differentiate it:
Substituting back gives:
Compare the coefficients, we have
Solve for A and B:
So the particular solution is
Now find the general solution to the homogenous equation
The roots of its characteristic equation are:
Then general solution to the homogenous ODE is
Therefore, the general solution to the given ODE is:
Now find the constants c1 and c2 using the initial values:
Differentiate y(x):
Then
Thus the solution of this IVP is:
5. Solve the differential equation, y is a function of x
So this DE can be solved by separation of variables.
Integrate on both sides:
Then
Substitute 1 for x to find constant A:
So the solution is
© BrainMass Inc. brainmass.com December 24, 2021, 10:00 pm ad1c9bdddf>https://brainmass.com/math/calculus-and-analysis/solving-differential-equations-433751