Solve: 𝐚 (d^2 y)/( d x^2)+6 (d y)/( d x)+9 y=0 b 10 (d^2 y)/( d x^2)+50 (d y)/( d x)+250 y=0 𝐜-(

step 1 Question number 6, d square y over d x square minus 6, dy over d x plus 10y is equal to 20 minus

Solve: 𝐚 (d^2 y)/( d x^2)+6 (d y)/( d x)+9 y=0 b 10 (d^2...

Solve: $\mathbf{a} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+6 \frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0$ b $10 \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+50 \frac{\mathrm{d} y}{\mathrm{~d} x}+250 y=0$ $\mathbf{c}-\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-3 \frac{\mathrm{d} y}{\mathrm{~d} x}+8 y=0$

Solve:
$\mathbf{a} \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+6 \frac{\mathrm{d} y}{\mathrm{~d} x}+9 y=0$
b $10 \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+50 \frac{\mathrm{d} y}{\mathrm{~d} x}+250 y=0$
$\mathbf{c}-\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-3 \frac{\mathrm{d} y}{\mathrm{~d} x}+8 y=0$

Key Concepts

Roots of the Characteristic Equation

The roots of the characteristic polynomial indicate the behavior of the solution. Distinct real roots lead to solutions that are simple exponentials, repeated roots require multiplying by an increasing power of x for each repetition, and complex roots result in solutions that involve sine and cosine functions, reflecting oscillatory behavior.

Characteristic Equation

The characteristic equation is derived by assuming a trial solution of the form e^(rx) for a linear homogeneous differential equation with constant coefficients. Substituting this trial solution into the differential equation yields a polynomial (the characteristic equation) in r, whose roots determine the nature of the general solution.

Constant Coefficients

When a differential equation has constant coefficients, it means that the numerical coefficients multiplying the derivatives are constant values. This simplifies the solution process because it allows the use of exponential functions to explore solutions, leading to a characteristic equation.

Homogeneous Linear Differential Equations

A homogeneous linear differential equation is one in which every term is a function of the dependent variable or its derivatives, with no free (non-homogeneous) term present. This property allows the principle of superposition to apply, meaning the sum of any two solutions is also a solution.

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