Why does the presenter leave the answer implicit instead of solving for y?
Conditions
- The solution contains multiple y terms.
- An explicit solution for y is not attempted.
Reasoning, step by step
- Observe the final implicit equation.
- Note the presence of multiple y terms ( and -xy).
- Conclude that solving for y explicitly would be difficult.
- Leave the answer in implicit form.
Example
The final solution is -x^ - xy = C. Because of the and -xy terms, the presenter states, 'I think we'll probably leave this and not solve this directly for y.'
Common misconceptions
- Assuming the final answer must always be solved explicitly for y.
- Thinking that implicit solutions are incorrect or incomplete.
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Related questions
The final implicit solution for the example equation (-3x^2 - y) is -x^ - xy = C.
Conditions: The equation is exact.; The terms are combined correctly without double-counting.
The term -xy appears in both the integration of M with respect to x and the integration of N with respect to y. When combining the results to form the potential function, shared terms should only be included once to avoid double-counting.
Conditions: The term -xy is present in both antiderivatives.
When integrating a multivariable function with respect to one variable, the other variables are treated as constants. Therefore, the 'constant' of integration can be any function that depends only on the other variables, which in this case is y.
Conditions: The integration is with respect to x.; The function M depends on both x and y.
To check if a differential equation is exact, you must verify that the partial derivative of the coefficient of dx with respect to y is equal to the partial derivative of the coefficient of dy with respect to x. If these mixed partial derivatives are equal, the equation is exact.
Conditions: The equation must be in the form .; M and N must have continuous first-order partial derivatives.
The steps involve finding the potential function by integrating M with respect to x and N with respect to y, then combining the results to form the implicit solution .
Conditions: The differential equation has been verified as exact.
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