S10

Classification of 2nd Order Linear ODEs

• General Form:

  • The general representation of a 2nd order linear ODE is given by:[ y'' + P(t)y' + Q(t)y = g(t) ]

  • Homogeneous vs Nonhomogeneous:

    • If ( g(t) = 0 ), the equation is homogeneous, otherwise it is nonhomogeneous.

  • Constant Coefficients:

    • If ( P(t) ) and ( Q(t) ) are constant functions, the equation has constant coefficients; otherwise, it has nonconstant coefficients.

Types of Differential Equations

1. Nonconstant Coefficients, Nonhomogeneous:

   • ( y'' + p(t)y' + q(t)y = g(t) )

2. Nonconstant Coefficients, Homogeneous:

   • ( y'' + p(t)y' + q(t)y = 0 )

3. Constant Coefficients, Nonhomogeneous:

   • ( y'' + ay' + by = g(t) )

4. Constant Coefficients, Homogeneous:

   • ( y'' + ay' + by = 0 )

• Remark:

  • Case IV is typically the easiest and was completely solved in the last lecture.

  • Case I remains the most general and challenging, having applications in special functions and quantum mechanics.

Wronskian and Linear Operators

• Notation:

  • Define differential operator:[ L[y] = y'' + p(t)y' + q(t)y ]

• The Wronskian:

  • Wronskian of two functions ( y_1 ) and ( y_2 ) is given by:[ W[y_1, y_2] = y_1y_2' - y_1'y_2 ]

• Properties of Linear Operators:

  1. Linear operators maintain linearity:

     • ( L[af(t) + bg(t)] = aL[f(t)] + bL[g(t)] )

  2. If two functions satisfy the linear equation, their linear combination also satisfies the equation.

The Principle of Linear Superposition

• Theorem (Superposition):

  • If ( y_1(t) ) and ( y_2(t) ) are solutions of the ODE ( L[y] = 0 ), then the linear combination ( C_1 y_1(t) + C_2 y_2(t) ) is also a solution for any constants ( C_1 ) and ( C_2 ).

• Proof involves demonstrating that the operator ( L ) applied to this linear combination yields zero.

General Solution of ODE

• To satisfy initial conditions, the general solution has the form:[ y(t) = C_1 y_1(t) + C_2 y_2(t) ]

• Determining Coefficients:

  • By substituting initial conditions:[ y(t_0) = y_0, , y'(t_0) = y_1 ]

  • Can solve for constants uniquely if the Wronskian is not zero.

Existence and Uniqueness Theorem

• Consider the initial value problem (IVP):[ y'' + P(t)y' + Q(t)y = g(t) ] with continuous functions on interval ( I ) containing ( t_0 ).

• Theorems ensure solutions exist and are unique:

  1. The IVP has a solution.

  2. The solution is unique within the interval.

  3. Solutions exist on the largest interval possible where the functions are continuous.

Example of Existence and Uniqueness

• Given IVP:[ (t^2-3)y'' + ty' - (t+3)y = 0 ] with conditions ( y(1) = 2 ) and ( y'(1) = 1 ).

• Identify intervals with potential discontinuity where the coefficients are not continuous.

• The largest open interval containing ( t_0 ) where the equation is continuously defined is ( (0, 3) ).

Monomial Solutions Exercise

• Problem Statement: Find two monomial solutions of the equation:[ 2y'' + 3y' - y = 0 ]

• Check if they form a fundamental set of solutions and find the general solution.

• Guessing a solution of the form ( y(t) = t^n ) leads to the characteristic equation to derive the roots and corresponding functions for the general solution.

Conclusion

• Fundamental Set of Solutions:

  • The solution set provides a means to express general solutions of 2nd order linear ODEs.

  • Solutions are linear combinations of independent solutions determined by the Wronskian.