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darryw
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Homework Statement
y' = (2x) / y+(x^2)y y(0) = -2
i realize this is sort of algebra prob, but i can't seem to separate x's from y's..
darryw said:Homework Statement
y' = (2x) / y+(x^2)y y(0) = -2
i realize this is sort of algebra prob, but i can't seem to separate x's from y's..
Homework Equations
The Attempt at a Solution
Mistake in line above. It looks like you are trying to say that 2x/(1 + x^2) = 2x/1 + 2x/x^2.darryw said:Please check my work thanks alot..
find solution in explicit form:
y' = (2x) / y+(x^2)y initial conditions: y(0) = -2
dy/dx = 2x/y(1+x^2)
y dy = 2x + (2/x) dx
darryw said:integrate both sides...
(1/2)y^2 = x^2 + 2ln|x| + c
y^2 = 2x^2 + 4ln|x| + 2c
y = root [2x^2 + 4ln|x| + c]
apply IC:
-2 = root [2(0)^2 + 4ln|0| + c]
-2 = (root 4) + c
c = -4 (i am reasong that root c is same as c because whether its root c or just c, both are still constants.. right?)
so my solution is:
y = root [2x^2 + 4ln|x| -4]
A differential equation is an equation that relates a function or set of functions to its derivatives. It expresses how a change in the dependent variable is related to a change in the independent variable.
Separating a differential equation can be difficult because it involves isolating the dependent and independent variables on opposite sides of the equation, which can be complex and involve multiple steps. Additionally, some differential equations may not have a solution that can be easily separated.
The purpose of separating a differential equation is to make it easier to solve. Once the equation is separated, each side can be integrated separately, making it easier to find the solution to the original equation.
One strategy is to multiply both sides of the equation by an integrating factor, which can help in isolating the dependent and independent variables. Another strategy is to use a substitution method, where a new variable is introduced to simplify the equation.
No, not all differential equations can be separated. Some equations may require more advanced techniques or may not have a solution that can be easily separated. It is important to consider the specific properties and characteristics of the equation before attempting to separate it.