Question

Graphs of some members of a family of solutions for a first-order differential equation $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\mathbf{f}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$are shown in Figure. The graphs of two implicit solutions, one that passes through the point (1, 21) and one that passes through (21, 3), are shown in blue. Reproduce the figure on a piece of paper. With coloured pencils trace out the solution curves for the solutions $\mathbf{y}\mathbf{=}{\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and $\mathbf{y}\mathbf{=}{\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$dfined by the implicit solutions such that ${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{1}$and ${\mathbf{y}}_{\mathbf{2}}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{=}\mathbf{3}$respectively. Estimate the intervals on which the solutions $\mathbf{y}\mathbf{=}{\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$and $\mathbf{y}\mathbf{=}{\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$are defined.

Short answer

A first order differential equation is an equation of the form F (ty, y) = 0. A solution of a first order differential equation is a function f(t) that makes F(t, f(t),f′(t))=0 for every value of t.

Step-by-step solution

finding the solutions for solving the sum by drawing graph

The solution that passes through (1, -1), that is, the solution $y_1$such that $y_1\left(1\right)=-1$would resemble the bule curve in the fourth quadrant. The solution is defined approximately on(0.7, 4.3)

finding the other solution;

The other solution the passes through (-1, 3) that is (-1)=3 would resemble the blue curve in the second quadrant. This solution is defined on$(-\infty ,0)$

finding the final answer by further solution

The intervals on which the solution$y=y_1\left(x\right)$ and$y=y_2\left(x\right)$ are defined are (0.7, 4.3) and $(-\infty ,0)$.