devyy The differential equation given is: dy/dx = xe^y + 2x This is a first-order ordinary differential equation (ODE) involving the function y and its derivative with respect to x. To analyze and solve this equation, we can rewrite it as: dy/dx – 2x = xe^y This form highlights that the left-hand side of the equation … Read more
To find a function f for the given vector field f = x i + y j + z k, we need to start by understanding what is meant by a function that satisfies f = f and that evaluates to zero at the origin, i.e., f(0, 0, 0) = 0. The expression f = … Read more
Finding the Value of x in the Equation 3x = 10 To solve for the value of x in the equation 3x = 10, we need to isolate x. Here’s a step-by-step breakdown of how to do that: Start with the original equation: 3x = 10 Isolate x: To do this, divide both sides of … Read more
Finding the Composition of Functions To find the composition of the functions f(x) = 9x^2 and g(x) = x^3, we follow these steps: Substitute g(x) into f(x): We need to compute f(g(x)), which means replacing every instance of x in f(x) with g(x). Given that g(x) = x^3, we can write: f(g(x)) = f(x^3) Substituting … Read more
Total Surface Area of a Pyramid To find the total surface area of a pyramid with a square base, we need to calculate the area of the base as well as the areas of the triangular faces. Step 1: Calculate the Area of the Base The base of our pyramid is a square with side … Read more
To find the partial derivatives dz/dx and dz/dy of the implicit function given by the equation x² + 4y² + 3z² = 1, we can follow these steps: Step 1: Differentiate the equation with respect to x Using implicit differentiation, we differentiate both sides of the equation with respect to x: d/dx (x²) + d/dx … Read more
Total Surface Area of Cubes The total surface area (TSA) of a cube is calculated using the formula: TSA = 6 * (side length)² Where the side length is the length of one edge of the cube. Let’s apply this formula to the cubes with the given side lengths: 1. Cube with side length 1 … Read more
To determine the probability of rolling a sum of 5 with two regular six-sided dice, we need to assess all the possible outcomes when the dice are rolled and how many of those outcomes result in a sum of 5. When rolling two six-sided dice, each die can land on a number between 1 and … Read more
To find how many positive integers less than 1000 are divisible by both 7 and 11, we first need to determine the least common multiple (LCM) of these two numbers. The LCM of 7 and 11 can be calculated because both are prime numbers. Hence, the LCM is simply their product: LCM(7, 11) = 7 … Read more
To solve the problem, let’s define the two numbers involved. Let x be the first number and y be the second number. According to the problem, we have the following relationships: One number (let’s say x) is eight more than twice another number (which is y): x = 2y + 8 The difference between the … Read more