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7-111.

What is the antiderivative of each of the following differential equations? Use implicit integration when necessary. Solve your equations for $y$. Be careful about introducing the constant of integration at the appropriate time. Homework Help ✎

1. $\frac { d y } { d x }= 7x + 3$

Notice that the derivative is in terms of $x$. Antidifferentiate!

1. $\frac { d y } { d x }= 7y + 3$

Compare and contrast part (a) with part (b).
Which one requires implicit integration?

1. $\frac { d y } { d x }= 7y^2$

Translation: The derivative is equal the the original function squared times a constant.
What can you conclude about the original function?
Is it linear, quadratic, cubic, exponential, trigonometric, etc.?
Use implicit integration to check your conjecture.

1. $\frac { d y } { d x } = 7$

Translation: The derivative of this function is a constant.
What can you conclude about the function?

1. $\frac { d y } { d x }= e^y$

$y = −\ln|−x + C|$
Explain the significance of the absolute value.