Given that k is an arbitrary positive constant, show that y2+kx2=9k is a general solution of the differential equation dxdy=9−x2−xy,∣x∣≤3.
I have been able to do this by making a substitution, but trying to do it using partial fractions doesn't work. Can someone tell me what I've done wrong?
Given that k is an arbitrary positive constant, show that y2+kx2=9k is a general solution of the differential equation dxdy=9−x2−xy,∣x∣≤3.
I have been able to do this by making a substitution, but trying to do it using partial fractions doesn't work. Can someone tell me what I've done wrong?
∫y1dy=∫x−9xdx
∫y1dy=21∫x+31+21∫x−31
lny2=ln∣(x+3)(x−3)k∣
y2=kx2−9k2
Which is not what I'm trying to achieve.
As |x|<3 ln(∣(x+3)(x−3)∣)+lnk=ln(∣x+3∣∣x−3∣)+lnk=ln(3+x)(3−x)k
Given that k is an arbitrary positive constant, show that y2+kx2=9k is a general solution of the differential equation dxdy=9−x2−xy,∣x∣≤3.
I have been able to do this by making a substitution, but trying to do it using partial fractions doesn't work. Can someone tell me what I've done wrong?
∫y1dy=∫x−9xdx
how did you get to that first step? surely
∫y1dy=−∫9−x2xdx ∫y1dy=∫x2−9xdx
although you have worked on from that, so maybe a typo?
Your 1st answer is mostly right. The k^2 term should just be k. Remember k is an arbitrary constant. That means you can just replace it with -k, or in fact whatever you like. Replacing with -k sets you straight