91Ó°ÊÓ

Use Laplace transforms to solve the initial value problems. \(x^{(3)}+x^{\prime \prime}-6 x^{\prime}=0 ; x(0)=0, x^{\prime}(0)=x^{\prime \prime}(0)=1\)

In Problems 31 through 35, the values of mass \(m\), spring constant \(k\), dashpot resistance \(c\), and force \(f(t)\) are given for \(a\) mass-spring-dashpot system with external forcing function. Solve the initial value problem $$ m x^{\prime \prime}+c x^{\prime}+k x=f(t), \quad x(0)=x^{\prime}(0)=0 $$ and construct the graph of the position function \(x(t) .\) \(m=1, k=4, c=0 ; f(t)=1\) if \(0 \leqq t<\pi, f(t)=0\) if \(t \geqq \pi\)

In Problems, transform the given differential equation to find a nontrivial solution such that \(x(0)=0 .\) $$ t x^{\prime \prime}+2(t-1) x^{\prime}-2 x=0 $$

Use Laplace transforms to solve the initial value problems. \(x^{(4)}-x=0 ; x(0)=1, x^{\prime}(0)=x^{\prime \prime}(0)=x^{(3)}(0)=0\)

In Problems 31 through 35, the values of mass \(m\), spring constant \(k\), dashpot resistance \(c\), and force \(f(t)\) are given for \(a\) mass-spring-dashpot system with external forcing function. Solve the initial value problem $$ m x^{\prime \prime}+c x^{\prime}+k x=f(t), \quad x(0)=x^{\prime}(0)=0 $$ and construct the graph of the position function \(x(t) .\) \(m=1, k=4, c=5 ; f(t)=1\) if \(0 \leqq t<2, f(t)=0\) if \(t \geqq 2\)

In Problems, transform the given differential equation to find a nontrivial solution such that \(x(0)=0 .\) $$ t x^{\prime \prime}-2 x^{\prime}+t x=0 $$

In Problems 31 through 35, the values of mass \(m\), spring constant \(k\), dashpot resistance \(c\), and force \(f(t)\) are given for \(a\) mass-spring-dashpot system with external forcing function. Solve the initial value problem $$ m x^{\prime \prime}+c x^{\prime}+k x=f(t), \quad x(0)=x^{\prime}(0)=0 $$ and construct the graph of the position function \(x(t) .\) \(m=1, k=9, c=0 ; f(t)=\sin t\) if \(0 \leqq t \leqq 2 \pi, f(t)=0\) if \(t>2 \pi\)

Use Laplace transforms to solve the initial value problems. \(x^{(4)}+x=0 ; x(0)=x^{\prime}(0)=x^{\prime \prime}(0)=0, x^{(3)}(0)=1\)

Use Laplace transforms to solve the initial value problems. \(x^{(4)}+13 x^{\prime \prime}+36 x=0 ; x(0)=x^{\prime \prime}(0)=0, x^{\prime}(0)=2\), \(x^{(3)}(0)=-13\)

In Problems 31 through 35, the values of mass \(m\), spring constant \(k\), dashpot resistance \(c\), and force \(f(t)\) are given for \(a\) mass-spring-dashpot system with external forcing function. Solve the initial value problem $$ m x^{\prime \prime}+c x^{\prime}+k x=f(t), \quad x(0)=x^{\prime}(0)=0 $$ and construct the graph of the position function \(x(t) .\) \(m=1, k=1, c=0 ; f(t)=t\) if \(0 \leqq t<1, f(t)=0\) if \(t \geqq 1\)