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Magazine Problem 6
Find the general solution of each of the differential equations in exercise. $$ \frac{d^{3} y}{d x^{3}}-6 \frac{d^{2} y}{d x^{2}}+5 \frac{d y}{d x}+12 y=0 $$
Problem 7
Consider the differential equation
$$
\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0
$$
(a) Show that \(e^{2 x}\) and \(e^{3 x}\) are linearly independent solutions of
this equation on the interval \(-\infty
Problem 7
Find the general solution of each of the differential equations in exercise. $$ \frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x}+16 y=0 $$
Problem 8
Find the general solution of each of the differential equations in exercise. $$ 4 \frac{d^{2} y}{d x^{2}}+4 \frac{d y}{d x}+y=0 $$
Problem 8
Consider the differential equation
$$
\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+y=0
$$
(a) Show that \(e^{x}\) and \(x e^{x}\) are linearly independent solutions of this
equation on the interval \(-\infty
Problem 9
Find the general solution of each of the differential equations in exercise. $$ \frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}+13 y=0 $$
Problem 9
Consider the differential equation
$$
x^{2} \frac{d^{2} y}{d x^{2}}-2 x \frac{d y}{d x}+2 y=0
$$
(a) Show that \(x\) and \(x^{2}\) are linearly independent solutions of this
equation on the interval \(0
Problem 10
Find the general solution of each of the differential equations in exercise. $$ \frac{d^{2} y}{d x^{2}}+6 \frac{d y}{d x}+25 y=0 $$
Problem 10
Consider the differential equation
$$
x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}-4 y=0
$$
(a) Show that \(x^{2}\) and \(1 / x^{2}\) are linearly independent solutions of
this equation on the interval \(0
Problem 11
Consider the differential equation
$$
\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+4 y=0
$$
(a) Show that each of the functions \(e^{x}, e^{4 x}\), and \(2 e^{x}-3 e^{4 x}\)
is a solution of this equation on the interval \(-\infty
