Show that y=C e^r x is a solution of y^''+a y^'+b y=0 if and only if r is a root of P(r)=r^2+a r

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Show that y=C e^r x is a solution of y^''+a y^'+b y=0 if and...

Show that $y=C e^{r x}$ is a solution of $y^{\prime \prime}+a y^{\prime}+b y=0$ if and only if $r$ is a root of $P(r)=r^{2}+a r+b .$ Then verify directly that $y=C_{1} e^{3 x}+C_{2} e^{-x}$ is a solution of $y^{\prime \prime}-2 y^{\prime}-3 y=0$ for any constants $C_{1}, C_{2} .$

Key Concepts

Exponential Function as a Candidate Solution

The use of an exponential function y = Ce^(rx) as a candidate solution for differential equations is based on its property that the derivatives of e^(rx) result in a factor of r each time. This form simplifies the differential equation to an algebraic equation by canceling the exponential term, thus proving that the exponential form is a valid approach to find the solution when r satisfies the characteristic equation.

Direct Verification by Substitution

This method involves calculating the derivatives of the proposed solution and substituting them back into the original differential equation. The process confirms that the function satisfies the equation for any constants present. It is a key step in validating that a particular solution, or a general solution expressed as a linear combination of exponential functions, indeed meets the requirements of the differential equation.

Homogeneous Linear Differential Equations with Constant Coefficients

These are differential equations where all terms are linear in the unknown function and its derivatives, and the coefficients of these terms are constants. Their solutions can often be expressed in terms of exponential functions because the differentiation of an exponential function yields another exponential multiplied by a constant, which aligns well with the structure of the equation.

Characteristic Equation

This is an algebraic equation derived by substituting an exponential trial solution, such as y = Ce^(rx), into the differential equation. The characteristic equation links the eigenvalue r with the coefficients of the differential equation (e.g., r^2 + ar + b = 0) and determines the behavior and form of the solution to the differential equation.

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