3.7 - Solved Examples
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Practice Questions
Test your understanding with targeted questions
Solve the equation d²y/dx² + 3dy/dx + 2y = 0 with initial conditions y(0)=1, y′(0)=0.
💡 Hint: Use the characteristic equation to find the roots.
Identify the roots of the characteristic equation from d²y/dx² + y = 0.
💡 Hint: Use the quadratic formula to identify the nature of roots.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What form does the general solution take for distinct real roots?
💡 Hint: Remember the scenario with distinct roots yields this format.
True or False: The roots of the characteristic equation dictate the form of the general solution.
💡 Hint: Consider what the roots indicate about the solution types.
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Challenge Problems
Push your limits with advanced challenges
Prove that the differential equation d²y/dx²−5dy/dx+6y=0 leads to a general solution embodying distinct roots when initial conditions are set to y(0)=4 and y′(0)=3.
💡 Hint: Start with the characteristic equation and follow through with initial conditions.
For the equation d²y/dx² + 6dy/dx + 9y=0, explore how repeated roots influence the solution's composition with initial values y(0)=1 and y′(0)=0.
💡 Hint: Recognize that repeated roots change the solution structure considerably.
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