Question

Answer question 1 only, please and thank you.

In Problems 1 through 10, find a function y = f(x) satisfy- ing the given differential equation and the prescribed initial condition. dy 1. = 2x + 1; y(0) =3 dx

Answer 1

`y=x^2+x+3`

Explanation:

Solve the given initial-value problem.

`(dy)/(dx) =2x+1; \ y(0)=3`

This is a separable differential equation. We can solve these as follows.

`\boxed{\left\begin{array}{ccc}\text{\underline{Separable Differential Equation:}}\\(dy)/(dx) =f(x)g(y)\\\\\rightarrow\int(dy)/(g(y))=\int f(x)dx \end{array}\right }`

`(dy)/(dx) =2x+1\\\\\Longrightarrow dy=(2x+1)dx\\\\\Longrightarrow \int dy=\int(2x+1)dx\\\\\Longrightarrow \boxed{y=x^2+x+C}`

Use the initial condition to find the arbitrary constant "C."

`y=x^2+x+C; \ \text{Recall} \ y(0)=3\\\\\Longrightarrow3=(0)^2+0+C\\\\\Longrightarrow \boxed{C=3}\\\\\therefore \boxed{\boxed{y=x^2+x+3}}`

Thus, the problem is solved.