Answer to Question #211916 in Differential Equations for Faith

Question

1.solve the following differential equation by using separation of variables method:

1.1.x(dy/dx)=4y.

2.show whether or not the following differential equations are separable:

2.1.dy/dx=t(in(s^2t))+8t².

2.2.dy/dx=ye^x+y/x²+2.

2.3.dy/dx=x+1/y-1.

Answer 1

Question 1:

Solve the following differential equation by using seperation of variables method:

1.1 x$\frac{dy}{dx}=4y$

Solution:

The equation can be written as

$\frac{dy}{4y}=\frac{dx}{x}$

$\int\frac{dy}{4y}=\int\frac{dx}{x}$

$\frac{ln(|y|)}{4}={ln(|x|)}+C$

Answer

$\frac{ln(|y|)}{4}=ln(|x|)+C$

Question 2

Show whether or not the following differential equations are separable.

2.1.dy/dx=t(in(s^2t))+8t²

2.2.dy/dx=ye^x+y/x²+2

2.3.dy/dx=x+1/y-1

Solution;

2.1

$\frac {dy}{dx}=t(in(s^2t))+8t^2$

The equation is inseparable.

It's a parametric equations. The variables of integration do not match those of the equation.

2.2

$\frac{dy}{dx}=ye^x+\frac{y}{x^2}+2$

$\frac{dy}{dx}=y(e^x+\frac{1}{x^2})+2$

$\frac{dy}{y}=((e^x+\frac{1}{x^2})+\frac{2}{y}))dx$

The equation is inseparable.

2.3

$\frac{dy}{dx}=\frac{x+1}{y-1}$

(y-1)dy=(x+1)dx

The equation is separable.