A triangle is scalene if the side lengths are different. if two sides have equal lengths, the triangle is isosceles. if all sides have equal lengths, the triangle is equilateral
Answer:
y_c = 2 + 10*x
Step-by-step explanation:
Given:
y'' = 0
Find:
- The solution to ODE such that y(0) = 2, y'(0) = 10
Solution:
- Assuming a solution y = Ce^(mt)
So, y' = C*me^(mt)
y'' = C*m^2e^(mt)
- Back substitute into given ODE, we get:
y'' = C*m^2e^(mt) = 0
e^(mt) can not be equal to zero
- Hence, m^2 = 0
m = 0 , 0 - (repeated roots)
- The complimentary function for repeated roots is:
y_c = (C1 + C2*x)*e^(m*t)
y_c = C1 + C2*x
- Evaluate @ y(0) = 2
2 = C1 + C2*0
C1 = 2
-Evaluate @ y'(0) = 10
y'(t) = C2 = 10
Hence, y_c = 2 + 10*x
Answer:
Step-by-step explanation:
Let
V
be the number of vertices of a polyhedron,
F
the number of faces of that polyhedron, and
E
be the number of edges. The quantity
χ
=
V
−
E
+
F
is called the Euler characteristic (of a polyhedron). In the case of convex polyhedra,
χ
=
2
.
Consider, for example, a tetrahedron (which is the simplest solid). It has 4 faces,
1
2
(
4
)
(
3
)
=
6
edges, and
1
3
(
4
)
(
3
)
=
4
vertices. Thus we have
V
−
E
+
F
=
4
−
6
+
4
=
2
.
Euler's formula holds for all Platonic solids (tetrahedron, cube, octahedron, dodecahedron, and icosahedron). Since a cube and an octahedron are dual polyhedra (each is formed by connecting the centers of the faces of the other), their
V
and
F
values are equal to the
F
an
V
values of the other. (The same is true for the dodecahedron and icosahedron).