How to do you solve these?
1) find an explicit general solution of the differential equation:
2) Find an implicit general solution of the differential equation:
y^(3) dy/dx= (y^(4)+1) cosx
3) Find an explicit particular solution of the initial value problem:
x dy/dx - y= 2x^(2) y, y(1)= 1
4) Find an explicit particular solution of the initial value problem:
dy/dx= 6e^(2x-y), y(0)= 0
5) A pitcher of buttermilk initially at 25 Degree celcius is to be cooled by setting it on the front porch, where the temperature is 0 degree celcius. Suppose that the temperature of the buttermilk has dropped to 15 Degree Celcius after 20 minutes, when will it be 5 degree celcius?
For the LHS we may substitute ==> , so
<-- Just for the record,
Again, we may drop the absolute value bars, but we must require that never be negative. But this is true anyway, so no worries!
Since all the question asked for is an implicit relationship, you can be done with this line here.
I forget whose name is attached to this equation, but we may simplify it somewhat by the substitution
So the differential equation becomes:
The characteristic equation for the homogeneous equation is
Thus m = 1 ==>
The particular solution is provided by a term of the form . Substitution in the t version of the differential equation gives:
Thus c = 2 (by matching the exponents) and Bc - B = 2 (by matching the coefficients) thus B = 2 also.
So the solution to the t version of the differential equation is
Now we need to revert back to the x variable. , so
Now we have the initial condition: .
and the solution to the differential equation will be:
So the temperature starts as and cools at a constant . How long will it take to drop to ?
Let me put it this way:
You are looking at a plane of graph paper where time, t, is on the x-axis and temperature, T, is on the y-axis. We have two points on the function we wish to graph:
I told you the function connecting these points is linear, that is to say, a line. I found the slope of this line, . Now you need to supply the following coordinate: What is the t value when the T value is ? ie. Fill in the following t coordinate: .
If you were asked this about the xy plane and a line connecting the points I am sure you would have no problem doing this.
I should note that in writing this I realized that is negative. I have fixed this in my original post on this question. Sorry for the error!