Differential Equations CALCULATOR Technique | Board Exam Approach (All types) | Most effective

Introduction

This tutorial provides a step-by-step approach to solving different types of differential equations using a calculator, as discussed in the video by Engr. Jhonard. The techniques covered will help you save time during board exams and enhance your understanding of differential equations, including variable separable, exact, inexact, and homogeneous equations.

Step 1: Understanding Differential Equations

• Familiarize yourself with the types of differential equations:
  • Variable Separable DE: These can be separated into two distinct variables.
  • Exact DE: These can be expressed in a way that allows for a straightforward integration.
  • Inexact DE: These require manipulation to become exact.
  • Homogeneous DE: These have a specific form that can be simplified.

Step 2: Setting Up the Calculator

• Ensure your calculator is equipped for solving differential equations. You may need:

  • A scientific calculator with differential equation capabilities.
  • Software or online tools that provide differential equation solving features.

• Check the settings to ensure it is in the correct mode for calculus operations.

Step 3: Solving Variable Separable Equations

• Identify variables in the equation that can be separated.
• Rearrange the equation to isolate variables:
  • Example format: ( \frac{dy}{dx} = g(x)h(y) )
• Integrate both sides:
  • Perform integration using your calculator:

    ∫dy/h(y) = ∫g(x)dx
    

• Solve for ( y ) to find the solution.

Step 4: Solving Exact Differential Equations

• Check if the differential equation is exact by verifying:

  • If ( M(x, y) ) and ( N(x, y) ) are partial derivatives.

• Use the formula for finding the potential function:

  • Integrate ( M ) with respect to ( x ) and ( N ) with respect to ( y ).

• Utilize your calculator's integration features to perform these calculations.

Step 5: Handling Inexact Differential Equations

• Convert an inexact differential equation into an exact one:
  • Identify an integrating factor ( μ(x) ) or ( μ(y) ).
• Multiply the entire equation by the integrating factor.
• Recheck for exactness and then follow the steps for exact equations.

Step 6: Solving Homogeneous Differential Equations

• Recognize the standard form: ( \frac{dy}{dx} = f\left(\frac{y}{x}\right) ).
• Use substitution ( v = \frac{y}{x} ) to simplify the equation.
• Rearrange and solve using your calculator's capabilities:
  • Example substitution leads to ( y = vx ), then differentiate to find ( dy/dx ).

Conclusion

By following these steps, you can effectively utilize a calculator to solve various types of differential equations, saving time during exams. Practice with different equations to become comfortable using your calculator for these techniques. As you prepare for your board exams, ensure you understand each type of equation and the appropriate solving methods.