Higher-Order Homogeneous DEs - Examples & Applications

Conceptual Examples: Existence and Superposition

Existence and Uniqueness (Conceptual)

Determine the largest interval II for which the Existence and Uniqueness Theorem guarantees a unique solution for the initial value problem:

(x2−4)y′′+3xy′−y=0,y(1)=4,y′(1)=−2(x^2 - 4)y'' + 3xy' - y = 0, \quad y(1) = 4, \quad y'(1) = -2

Existence and Uniqueness Solution

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Superposition Principle (Conceptual)

You are given that y1=exy_1 = e^x and y2=e−xy_2 = e^{-x} are both solutions to the homogeneous differential equation y′′−y=0y'' - y = 0. Using the Superposition Principle, find a solution that satisfies the boundary conditions y(0)=5y(0) = 5 and y(ln⁡2)=3y(\ln 2) = 3.

Superposition Principle Solution

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Linear Independence and the Wronskian

Wronskian Independence Check (Basic)

Check if y1=sin⁡(x)y_1 = \sin(x) and y2=cos⁡(x)y_2 = \cos(x) are linearly independent.

Wronskian Test Solution

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Wronskian Check (Intermediate)

Determine if the functions y1=x2y_1 = x^2 and y2=x2ln⁡xy_2 = x^2 \ln x are linearly independent on the interval (0,∞)(0, \infty).

Wronskian Solution 2

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Constant Coefficients

Homogeneous DE (Distinct Real Roots)

Find the general solution of the differential equation:

y′′−5y′+6y=0y'' - 5y' + 6y = 0

Distinct Roots Solution

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Homogeneous DE (Repeated Real Roots)

Solve the initial value problem:

y′′−4y′+4y=0,y(0)=1,y′(0)=0y'' - 4y' + 4y = 0, \quad y(0) = 1, \quad y'(0) = 0

Repeated Roots Solution

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Homogeneous DE (Complex Roots)

Solve the differential equation:

y′′+2y′+5y=0y'' + 2y' + 5y = 0

Complex Roots Solution

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Higher-Order Constant Coefficients

Find the general solution of the third-order equation:

y′′′−3y′′+3y′−y=0y''' - 3y'' + 3y' - y = 0

Higher-Order Solution

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Cauchy-Euler Equations

Cauchy-Euler Problem (Distinct Roots)

Solve the Cauchy-Euler equation:

x2y′′−2xy′+2y=0x^2y'' - 2xy' + 2y = 0

Cauchy-Euler Distinct Solution

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Cauchy-Euler Problem (Repeated Roots)

Solve the equation:

4x2y′′+8xy′+y=04x^2 y'' + 8x y' + y = 0

Cauchy-Euler Repeated Solution

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Cauchy-Euler Problem (Complex Roots)

Solve the equation:

x2y′′+xy′+4y=0x^2 y'' + x y' + 4y = 0

Cauchy-Euler Complex Solution

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