3 Solved Problems Variable Separable 2

Introduction to Variable Separable Differential Equations

In this section, the speaker introduces the formula for a variable separable differential equation and explains that integration is the only way to solve it.

Formula for a Variable Separable Differential Equation

• A variable separable differential equation has the form a(x,y)dx + b(x,y)dy = 0.

• Integration is necessary to solve this type of differential equation.

Converting a Variable Separable Differential Equation

In this section, the speaker explains how to convert a variable separable differential equation into an integrable form.

Converting with Respect to x and y

• To convert a variable separable differential equation with respect to x and y, divide both sides by one of the variables.

• If integrating with respect to x, divide by y. If integrating with respect to y, divide by x.

Example of Solving a Variable Separable Differential Equation

In this section, the speaker provides an example of solving a variable separable differential equation using integration.

Example Problem

• Given dx = t(1+t^2)(sec^2)x dt, separate variables and integrate.

• Divide both sides by (sec^2)x so that dx/(sec^2)x = t(1+t^2)dt.

• Integrate both sides. The left side becomes tanx + C. The right side becomes (1/2)t^2 + (1/4)t^4 + C.

• Multiply both sides by 4 so that 8tanx + 4t^2 + t^4 = C.

Solving a Variable Separable Differential Equation with Natural Logarithms

In this section, the speaker explains how to solve a variable separable differential equation that involves natural logarithms.

Example Problem

• Given yln(x)dy + dx = 0, separate variables and integrate.

• Divide both sides by yln(x) so that dy/y + dx/ln(x) = 0.

• Integrate the left side using integration by parts. The result is yln(y) - y + C.

• Substitute back in the original variables to get yln(xy) - xy + C = 0.

Solving Differential Equations

In this section, the speaker discusses how to solve differential equations using integration.

Separation of Variables Method

• The natural logarithm can be used to solve differential equations in the form of dy/dx = f(x)/g(y).

• The speaker provides an example of solving a differential equation using separation of variables method.

• After separating the variables, we can integrate both sides and solve for y.

Factoring Method

• The speaker provides another example of solving a differential equation using factoring method.

• After factoring out common terms, we can separate the variables and integrate both sides.

Final Answer

• Once we have integrated both sides, we can solve for y or x depending on which variable is isolated.

• We must add a constant (C) to our final answer since indefinite integrals have an arbitrary constant.

• The final answer will be in terms of x or y depending on which variable was isolated during integration.