3.7. Solved Examples

Try these first

1. 1.

   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.

2. 2.

   Identify the roots of the characteristic equation from d²y/dx² + y = 0.

   Hint

   Use the quadratic formula to identify the nature of roots.

3. 3.

   What form does the general solution take for distinct real roots?

   • y = C₁e^(r₁x) + C₂e^(r₂x)
   • y = (C₁ + C₂x)e^(r₁x)
   • y = e^(αx)(C₁cos(βx) + C₂sin(βx))

   Hint

   Remember the scenario with distinct roots yields this format.

4. 4.

   True or False: The roots of the characteristic equation dictate the form of the general solution.

   • True
   • False

   Hint

   Consider what the roots indicate about the solution types.

5. 5.

   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.

6. 6.

   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.