First Order Linear Differential Equation, 2.3#29

TL;DR

Learn how to solve a differential equation with e as the base by reversing the roles of the independent and dependent variables.

Transcript

let's see how can solve this differential equation we have dy/dx is equal to 1 over e to e for y and then we have the plus 2x after that hmm if I multiply this on both sides and try to separate the variables it's not going to work because we have to eat we need more y plus 2 it I cannot factor all the Y's together all the eggs together you can try ... Read More

Key Insights

• ⚾ Reversing the roles of the independent and dependent variables helps in solving differential equations with e as the base.
• ✊ The integrating factor is determined by taking the integral of P(Y) and raising e to the power of the integral.
• 🙃 The solution is obtained by integrating both sides of the equation and dividing by e to the power of negative 2Y.

Questions & Answers

Q: How is the differential equation transformed into a linear form?

The differential equation is transformed by reversing the roles of the independent and dependent variables. This is done by taking the reciprocal of the original equation, resulting in DX/dY on the left-hand side and the equation with P(Y) on the right-hand side.

Q: What is the integrating factor used for?

The integrating factor is used to simplify the equation and find the solution. It is determined by taking the integral of P(Y) and multiplying it with the equation. This helps in transforming the equation into a linear form.

Q: How is the integrating factor calculated?

The integrating factor is calculated by taking the integral of P(Y) and raising e to the power of the integral. In this case, the integral is negative 2 times dy, resulting in e to the power of negative 2Y.

Q: How is the solution obtained?

After multiplying the original equation with the integrating factor, the left-hand side becomes the derivative of X with respect to Y. By integrating both sides, the solution can be found. The final solution for X is obtained by dividing everything by e to the power of negative 2Y.

Summary & Key Takeaways

• The video discusses how to solve a differential equation with the base e by reversing the roles of the independent and dependent variables.

• By using the integrating factor, the equation is transformed into a linear form with the dependent variable being X and the independent variable being Y.

• The integrating factor is determined by taking the integral of P(Y), which is then multiplied with the equation to simplify it and find the solution.