91Ó°ÊÓ

For each of the following differential equations: a. Solve the initial value problem. b. [T] Use a graphing utility to graph the particular solution. $$ y^{\prime \prime}+5 y^{\prime}+15 y=0 \quad y(0)=-2, \quad y^{\prime}(0)=7 $$

(Principle of superposition) Prove that if \(y_{1}(x)\) and \(y_{2}(x)\) are solutions to a linear homogeneous differential equation, \(y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0,\) then the function \(y(x)=c_{1} y_{1}(x)+c_{2} y_{2}(x), \quad\) where \(\quad c_{1}\) and \(c_{2}\) are constants, is also a solution.

Prove that if \(a, b,\) and \(c\) are positive constants, then all solutions to the second-order linear differential equation \(a y^{\prime \prime}+b y^{\prime}+c y=0\) approach zero as \(x \rightarrow \infty .\) (Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)

Solve the following equations using the method of undetermined coefficients. $$ 2 y^{\prime \prime}-5 y^{\prime}-12 y=6 $$

Solve the following equations using the method of undetermined coefficients. $$ 3 y^{\prime \prime}+y^{\prime}-4 y=8 $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}-6 y^{\prime}+5 y=e^{-x} $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}-4 y^{\prime}+4 y=8 x^{2}+4 x $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}-2 y^{\prime}-3 y=\sin 2 x $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}+2 y^{\prime}+y=\sin x+\cos x $$

Solve the following equations using the method of undetermined coefficients. $$ y^{\prime \prime}+9 y=e^{x} \cos x $$