Determining the value of the y-intercept
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Unknown008
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Do you know how to deal with differential equations?
Unknown008
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Ahh no. Okay, for the second equation:
\(\displaystyle \displaystyle \frac{dy_2}{dx} = 8x + 5\)
\(\displaystyle \displaystyle\int dy_2 = \int 8x + 5\ dx\)
Can you take it from here?
\(\displaystyle \displaystyle \frac{dy_2}{dx} = 8x + 5\)
\(\displaystyle \displaystyle\int dy_2 = \int 8x + 5\ dx\)
Can you take it from here?
Unknown008
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Right!
Okay, now, can you do the same for the first differential equations.
And yes, you do integration with differential equations.
Okay, now, can you do the same for the first differential equations.
And yes, you do integration with differential equations.
Unknown008
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No, this one is a little different.
\(\displaystyle \displaystyle \frac{dy_1}{dx} = 3y_1\)
\(\displaystyle \displaystyle \int \frac{1}{y_1} dy_1 = \int 3\ dx\)
Do you know how to integrate this one?
\(\displaystyle \displaystyle \frac{dy_1}{dx} = 3y_1\)
\(\displaystyle \displaystyle \int \frac{1}{y_1} dy_1 = \int 3\ dx\)
Do you know how to integrate this one?
Unknown008
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We'll get there.
You should first make sure that you have the two equations first.
Then, from the statement that they have the same y-intercept, you put in both equations, (0, y) [that is x = 0, and y = y]
Then, they intercept at x = 2 means that when x = 2, the y value is the same for both equations, that is substitute 2 and solve for y.
From those, you should be able to get the two different constants of integration.
You should first make sure that you have the two equations first.
Then, from the statement that they have the same y-intercept, you put in both equations, (0, y) [that is x = 0, and y = y]
Then, they intercept at x = 2 means that when x = 2, the y value is the same for both equations, that is substitute 2 and solve for y.
From those, you should be able to get the two different constants of integration.
Unknown008
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Don't forget the y part.
\(\displaystyle \displaystyle\int \frac{1}{y}\ dy = \ln(y) + c\)
Unknown008
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Not quite.
From the first differential equation:
\(\displaystyle \displaystyle \int \frac{1}{y_1} dy_1 = \int 3\ dx\)
This becomes:
\(\displaystyle \ln(y_1) = 3x + c_1\)
c_1 is the first constant.
From the second differential equation:
\(\displaystyle \displaystyle\int dy_2 = \int 8x + 5\ dx\)
This becomes:
\(\displaystyle y_2 = 4x^2 + 5x + c_2\)
Those are your two equations.
\(\displaystyle \ln(y_1) = 3x + c_1\)
\(\displaystyle y_2 = 4x^2 + 5x + c_2\)
From the first statement, put x = 0 and y1 = y2 = y and solve for both equations, to get an equation in terms of c1 and c2.
From the second statement, put x = 2 and y1 = y2 = y and solve for both equations, to get an equation in terms of c2 and c2 again.
With two equations and two unknowns, solve them simultaneously to get the value of the constants.
Then replace them back into the two equations to get the final answers.
From the first differential equation:
\(\displaystyle \displaystyle \int \frac{1}{y_1} dy_1 = \int 3\ dx\)
This becomes:
\(\displaystyle \ln(y_1) = 3x + c_1\)
c_1 is the first constant.
From the second differential equation:
\(\displaystyle \displaystyle\int dy_2 = \int 8x + 5\ dx\)
This becomes:
\(\displaystyle y_2 = 4x^2 + 5x + c_2\)
Those are your two equations.
\(\displaystyle \ln(y_1) = 3x + c_1\)
\(\displaystyle y_2 = 4x^2 + 5x + c_2\)
From the first statement, put x = 0 and y1 = y2 = y and solve for both equations, to get an equation in terms of c1 and c2.
From the second statement, put x = 2 and y1 = y2 = y and solve for both equations, to get an equation in terms of c2 and c2 again.
With two equations and two unknowns, solve them simultaneously to get the value of the constants.
Then replace them back into the two equations to get the final answers.
Unknown008
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Very Good! 
D
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Very Good!
I would like to add that you have done a fantastic job at teaching the student. You made the student find the answer - instead of spoon-feeding. It took a bit of time - but the student will have "learned" the procedure thoroughly.
Thanks for your patience.....