Find a particular solution from the differential equation to the following
I can't separate the variables, and also when I use the answers doesn't match the ans key which is
Same problem with this question
solve the DE :
What method do we use for this type of DE?
Okay, I tried it again, please check whether my answer is correct or not
laTexis not working properly, so I type it like this
\frac{1}{x} dx = \frac{2v}{1+ v^2} dv
integrateboth sides
ln|x|= ln | 1 + v^2|
x= 1 + v^2
x= 1 + \frac({y}{x})^2 + C
y(1)= 2 --> 1 = 1 + 4 + C
C= -4
x= 1 + v^2 - 4
v= \sqrt{x+3}
\frac{y}{x}= \sqrt{x+3}
y= x \sqrt{x+3}
y= \sqrt{x^3 + 3x^2}
No, that is not correct. The constant of integration comes when you integrate, not when you exponentiate. On line 4, you write "integrate both sides". The next line should read:
ln|x| + C = ln|1+v^2|
Note that if you add a constant to both sides, you can subtract it from the right and the constant on the left "absorbs" the one from the right.
Now, when you exponentiate, you get:
e^(ln|x| + C) = 1+v^2
e^(ln|x|)e^C = 1+v^2
xe^C = 1+v^2
Now, I tend to write K for multiplicative constants. Let K = e^C:
Kx = 1+v^2
Now, Kx = 1 + (\frac{y}{x})^2, so y(1) = 2 --> K = 1+(\frac{2}{1})^2 = 5.
5x = 1+v^2 implies v = \pm \sqrt{5x-1}
y = xv = \pm x\sqrt{5x-1} = \pm \sqrt{5x^3-x^2}