A more detailed version of Theorem 1 says that, if the function f(x, y) is continuous near the point (a, b), then at least one so- lution of the differential equation y' = f(x, y) exists on some open interval I containing the point x = a and, moreover, that if in addition the partial derivative af/dy is continuous near (a, b), then this solution is unique on some (perhaps smaller) interval J. In Problems 11 through 20, determine whether ex- istence of at least one solution of the given initial value prob- lem is thereby guaranteed and, if so, whether uniqueness of that solution is guaranteed. dy 11. = 2x2 y2; y(1) = -1 dx dy 12. = x In y; y(1) = 1 %3D dx dy 13. = Vy; y(0) = 1 %3D dx dy 14. dx Vy: y(0) = 0 %3D dy 15. = - y; y(2) = 2 dx dy 16. = -yi y(2) = 1 dx dy dx 17. y = x-1; y(0) = 1 dy 18. y = x-13; y(1) =0 %3D dy
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
Do 17 and 18.
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