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SunGod87
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Trying to solve the two questions attached, for the first one:
du/dx - x.du/dy = 0
Assume u = X.Y
Y.dX/dx - x.X.dY/dy = 0
Dividing by x.X.Y and taking one term over to the other side:
dX/dx.1/(x.X) = dY/dy.1/Y
These can be equated to a constant m
dX/X = m.x.dx
Integrating;
ln(X/C) = 1/2.m.x^2
X = C.e^((1/2.m.x^2))
dY/Y = m.dy
ln(Y/C) = m.y
Y = C.e^((y.m))
u = C^2.e^((1/2.x^2 + y).m)
However the answer given is C.e^((x^2 + 2y).m)
Have they absorbed the two constants earlier into one? Since the equation is a first order one. Also they seem to have multiplied the exponent by 2 throughout, surely this isn't allowed so I assume I've made a mistake?
Second question:
x.du.dx - 2y.du/dy = 0
Assume u = X.Y
x.Y.dX/dx - 2y.X.dY/dy = 0
Dividing by 2.X.Y and taking one term to the other side:
x/2X.dX/dx = y/Y.dY/dy
These can be equated to a constant m
y/Y.dY/dy = m
dY/Y = m.dy/y
ln(Y/C) = m.ln(y)
ln(Y/C) = ln(y^m)
Y = C.y^m
x/2X.dX/dx = m
dX/2X = m.dx/x
1/2.ln(X/C) = m.ln(x)
ln(X/C) = 2.m.ln(x)
ln(X/C) = ln(x^2m)
X = C.x^2m
XY = C^2.(x^2.y)^m
However the book has it as:
C(x^2.y)^m
Again have they just combined the two constants into one since the equation is only first order?
Thanks in advance for any help :)
du/dx - x.du/dy = 0
Assume u = X.Y
Y.dX/dx - x.X.dY/dy = 0
Dividing by x.X.Y and taking one term over to the other side:
dX/dx.1/(x.X) = dY/dy.1/Y
These can be equated to a constant m
dX/X = m.x.dx
Integrating;
ln(X/C) = 1/2.m.x^2
X = C.e^((1/2.m.x^2))
dY/Y = m.dy
ln(Y/C) = m.y
Y = C.e^((y.m))
u = C^2.e^((1/2.x^2 + y).m)
However the answer given is C.e^((x^2 + 2y).m)
Have they absorbed the two constants earlier into one? Since the equation is a first order one. Also they seem to have multiplied the exponent by 2 throughout, surely this isn't allowed so I assume I've made a mistake?
Second question:
x.du.dx - 2y.du/dy = 0
Assume u = X.Y
x.Y.dX/dx - 2y.X.dY/dy = 0
Dividing by 2.X.Y and taking one term to the other side:
x/2X.dX/dx = y/Y.dY/dy
These can be equated to a constant m
y/Y.dY/dy = m
dY/Y = m.dy/y
ln(Y/C) = m.ln(y)
ln(Y/C) = ln(y^m)
Y = C.y^m
x/2X.dX/dx = m
dX/2X = m.dx/x
1/2.ln(X/C) = m.ln(x)
ln(X/C) = 2.m.ln(x)
ln(X/C) = ln(x^2m)
X = C.x^2m
XY = C^2.(x^2.y)^m
However the book has it as:
C(x^2.y)^m
Again have they just combined the two constants into one since the equation is only first order?
Thanks in advance for any help :)
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