Hi,
I am new to differential equations so please go easy on me. I am stuck with a couple of questions for simple 1st order differential equations:
Q1) Find general solution to dy/dx = -x/y
Q2) Find general solution to dy/dx = - y/x and graph at least 2 solutions
My attempt:
Q1) y dy = -x dx (separate variables)
y2 / 2 + C1 = - x2 /2 + C2
y2 / 2 = - x2 /2 + C3 (where C3 = C2 - C1)
y = - x + C
I don't see what I have done wrong here but the solution apparently is circular solution (x2 + y2 = r2 where r2 = 2C) - I can't see anything invalid from what I have done?
Q2) 1/y dy = - 1/x dx
ln (y) = - ln(x) + C
ln (x) + ln (y) = C
ln (xy) = C
y = ec/x
I plotted graphs of e1 and e2 but I'm note sure what do graph when it is e1 or e2 divided by x? What would dividing by x do to a graph of e1 compared to plotting just e1?
Again this solution and graphing is also apparently wrong (but not as much as the last one) - the solution shows taking mod xy (|xy|) so the final solution shown is:
y = + eC / x
The graph for this solution shows hyperbolas symmetric about the y-axis which I understand since they have used the modulus. I really don't understand why they took the modulus? How am I supposed to know to take the modulus rather than what I did?
Any help offered is much appreciated!!!
I am new to differential equations so please go easy on me. I am stuck with a couple of questions for simple 1st order differential equations:
Q1) Find general solution to dy/dx = -x/y
Q2) Find general solution to dy/dx = - y/x and graph at least 2 solutions
My attempt:
Q1) y dy = -x dx (separate variables)
y2 / 2 + C1 = - x2 /2 + C2
y2 / 2 = - x2 /2 + C3 (where C3 = C2 - C1)
y = - x + C
I don't see what I have done wrong here but the solution apparently is circular solution (x2 + y2 = r2 where r2 = 2C) - I can't see anything invalid from what I have done?
Q2) 1/y dy = - 1/x dx
ln (y) = - ln(x) + C
ln (x) + ln (y) = C
ln (xy) = C
y = ec/x
I plotted graphs of e1 and e2 but I'm note sure what do graph when it is e1 or e2 divided by x? What would dividing by x do to a graph of e1 compared to plotting just e1?
Again this solution and graphing is also apparently wrong (but not as much as the last one) - the solution shows taking mod xy (|xy|) so the final solution shown is:
y = + eC / x
The graph for this solution shows hyperbolas symmetric about the y-axis which I understand since they have used the modulus. I really don't understand why they took the modulus? How am I supposed to know to take the modulus rather than what I did?
Any help offered is much appreciated!!!