K(x,y,p)= integral G dx (range upper d, bottom c)
To get rid of the integral, we differentiate both sides of the equation with respect to . On the left hand side of the resulting equation we obtain the following expression (which might involve x, y, p and p=dy/dx q=dp/dx =dy^2/dx^2
-xq/30
(remember to separate different variables in a product with spaces or multiplication signs)
while on the right hand side (after applying the Fundamental Theorem of Calculus) we obtain
-(1+p^2)^(1/2)/80
The resulting differential equation we obtained above is a separable differential equation in the variables and . We can rewrite it in the form
K(p)dp=L(x)dx
(with all numerical factors moved to the right hand side of the equation so that L(1)=30/80.) where
K(p)=
L(x)=
Integrating the left hand side of the equation integral K(p)dp, using the methods of sections 8.3 and 8.2, we obtain
Integral K(p)dp=
while on the right hand side we obtain integral L(x)dx= +C
Plugging in the initial positions of the hawk and pigeon, and recalling that p=dy/dx is the slope of the tangent line, we find that
C=
Note that it is important to compute to at least 5 or 6 decimal places of precision. While computing C to less precision may not affect the correctness of your answer at this point, the roundoff error tends to get magnified in your subsequent calculations in problems 5 and 6 and may lead to your answers there being unexpectedly rejected as incorrect.
Solving the equation we obtained in Problem 4 for P in terms of x, we obtain
P=
(Hints for solving for P: Exponentiate to get rid of the logarithm. Then isolate the square root on one side of the equation and square both sides.)
Recalling that P=dy/dx and integrating. we obtain that
Y= +c
Plugging in the initial position of the hawk we obtain that the constant of integration is given by
C =
Note: if your answer is unexpectedly rejected as incorrect, go back to problem 4 part c, recompute your constant of integration there to more decimal places and recalculate your answers here with this more accurate value.
Hence the hawk catches the pigeon at the point (0,c) where
C=
at time t =
To get rid of the integral, we differentiate both sides of the equation with respect to . On the left hand side of the resulting equation we obtain the following expression (which might involve x, y, p and p=dy/dx q=dp/dx =dy^2/dx^2
-xq/30
(remember to separate different variables in a product with spaces or multiplication signs)
while on the right hand side (after applying the Fundamental Theorem of Calculus) we obtain
-(1+p^2)^(1/2)/80
The resulting differential equation we obtained above is a separable differential equation in the variables and . We can rewrite it in the form
K(p)dp=L(x)dx
(with all numerical factors moved to the right hand side of the equation so that L(1)=30/80.) where
K(p)=
L(x)=
Integrating the left hand side of the equation integral K(p)dp, using the methods of sections 8.3 and 8.2, we obtain
Integral K(p)dp=
while on the right hand side we obtain integral L(x)dx= +C
Plugging in the initial positions of the hawk and pigeon, and recalling that p=dy/dx is the slope of the tangent line, we find that
C=
Note that it is important to compute to at least 5 or 6 decimal places of precision. While computing C to less precision may not affect the correctness of your answer at this point, the roundoff error tends to get magnified in your subsequent calculations in problems 5 and 6 and may lead to your answers there being unexpectedly rejected as incorrect.
Solving the equation we obtained in Problem 4 for P in terms of x, we obtain
P=
(Hints for solving for P: Exponentiate to get rid of the logarithm. Then isolate the square root on one side of the equation and square both sides.)
Recalling that P=dy/dx and integrating. we obtain that
Y= +c
Plugging in the initial position of the hawk we obtain that the constant of integration is given by
C =
Note: if your answer is unexpectedly rejected as incorrect, go back to problem 4 part c, recompute your constant of integration there to more decimal places and recalculate your answers here with this more accurate value.
Hence the hawk catches the pigeon at the point (0,c) where
C=
at time t =