Unlock the power of differential equations with our comprehensive guide. Discover step-by-step solutions to complex mathematical problems, including homogeneous and non-homogeneous equations, characteristic polynomials, and particular solutions. Learn how to apply these concepts to real-world scenarios and improve your understanding of calculus. Our detailed explanations and examples will help you master differential equations and enhance your problem-solving skills. Dive into the world of mathematics and explore the solutions to exercises 6 and 7, covering topics such as parametric representation, spherical coordinates, and critical points. Boost your knowledge and confidence in tackling challenging mathematical problems.
Contents
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
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Extract
[...] In : et so admits a relative maximum in . In : so admits a point col in . Exercise 2 So: . On a so the right may be represented by: . On a on the right can be represented by: . On a so the right may be represented by: . On a donc . So be it the normal vector to the plane passing through the points , and and departing from the octahedron. Then we have . [...]
[...] Finally we have and the general solution is written as: so . Then we have . so so so . On a then . Finally . If we limit ourselves to order 3 we finally get: . Exercice 6 So et finalement : On a donc the same form as with . Using the question we can replace the left-hand side so: D'where then and finally: or even: The derivative is worth . et so . The solution is therefore . [...]
[...] The general solution is therefore written as: . The general solution of the differential equation est : The general solution of the differential equation est : According to the solutions of the homogeneous equation are: . On cherche une solution particulière de l'équation différentielle complexe : under the form et . So . On a donc On chooses a cubic polynomial: then and so . so . (on prendra ) So . We have the particular solution of the real equation . [...]
[...] Method 1 (direct calculation of circulation): so . On a donc . Method 2 (Stokes' theorem): Exercice 4 On a so . It is a half-sphere of radius (demie only because This is the sphere of radius and center of coordinates . The surface is limited by the sphere . On reports the parametric representation in the equation of the sphere: The parameter so it will vary from to and of à . So we have: . Exercice 5 On cherche une solution générale de the homogeneous equation . [...]
