
New Member
dy/dx=x+y
Find y_{0 }so that the integral curve for dy/dx=x+y y(4)=y_{0} is a straight line. You must justify your answer, which will require you to apply algebraic reasoning to the problem.
I know the answer is 3. I also know this differential equation looks simple, but I can't get the starting equation separated. We are studying separable equations so I must do it using that method. If someone could help me get it separated I think I can handle it. Thank you.

Elite Member
I'm feeling an urge to do a change of variables: y = t*x

New Member
Firstly, the DE y' = x + y can be solved like such:
Multiply by e^{x}: e^{x}y'  e^{x}y = xe^{x
}
Integrate: e^{x}y = e^{x}(x+1) + c_{1}
Solve; divide by e^{x}: y(x) = c_{1}e^{x}  x  1
Secondly, solve for c_{1}.
IVP: y(4) = y_{0} = c_{1}e^{4} + 3
Solve: c_{1} = (y_{0}  3)e^{4}
Input: y(x) = (y_{0}  3)e^{x}^{+4}  x  1
When y_{0}  3, the equation is y(x) = x  1.
This can be reasoned as the only solution because while y = x  1 by definition a strait line, y=ce^{x}^{+4} is not a strait line except at c=0. There are, of course, other mathematical ways of proving this.
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