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Solve the following initial-value problem and graph the solution: $y\text{\u2033}-2{y}^{\prime}+10y=0,y(0)=2,{y}^{\prime}(0)=\mathrm{-1}$
$y(x)={e}^{x}(2\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}3x-\text{sin}\phantom{\rule{0.1em}{0ex}}3x)$
The following initial-value problem models the position of an object with mass attached to a spring. Spring-mass systems are examined in detail in Applications . The solution to the differential equation gives the position of the mass with respect to a neutral (equilibrium) position (in meters) at any given time. (Note that for spring-mass systems of this type, it is customary to define the downward direction as positive.)
Solve the initial-value problem and graph the solution. What is the position of the mass at time $t=2$ sec? How fast is the mass moving at time $t=1$ sec? In what direction?
In [link] c. we found the general solution to this differential equation to be
Then
When $t=0,$ we have $y(0)={c}_{1}$ and ${y}^{\prime}(0)=\text{\u2212}{c}_{1}+{c}_{2}.$ Applying the initial conditions, we obtain
Thus, ${c}_{1}=1,$ ${c}_{2}=1,$ and the solution to the initial value problem is
This solution is represented in the following graph. At time $t=2,$ the mass is at position $y(2)={e}^{\mathrm{-2}}+2{e}^{\mathrm{-2}}=3{e}^{\mathrm{-2}}\approx 0.406$ m below equilibrium.
To calculate the velocity at time $t=1,$ we need to find the derivative. We have $y(t)={e}^{\text{\u2212}t}+t{e}^{\text{\u2212}t},$ so
Then ${y}^{\prime}(1)=\text{\u2212}{e}^{\mathrm{-1}}\approx -0.3679.$ At time $t=1,$ the mass is moving upward at 0.3679 m/sec.
Suppose the following initial-value problem models the position (in feet) of a mass in a spring-mass system at any given time. Solve the initial-value problem and graph the solution. What is the position of the mass at time $t=0.3$ sec? How fast is it moving at time $t=0.1$ sec? In what direction?
$y(t)=t{e}^{\mathrm{-7}t}$
At time
$t=0.3,$
$y(0.3)=0.3{e}^{(\mathrm{-7}*0.3)}=0.3{e}^{\mathrm{-2.1}}\approx 0.0367.$ The mass is 0.0367 ft below equilibrium. At time
$t=0.1,$
${y}^{\prime}(0.1)=0.3{e}^{\mathrm{-0.7}}\approx 0.1490.$ The mass is moving downward at a speed of 0.1490 ft/sec.
In [link] f. we solved the differential equation $y\text{\u2033}+16y=0$ and found the general solution to be $y(t)={c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}4t+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}4t\text{.}$ If possible, solve the boundary-value problem if the boundary conditions are the following:
We have
Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous.
${x}^{3}y\text{\u2033}+\left(x-1\right){y}^{\prime}-8y=0$
linear, homogenous
$\left(1+{y}^{2}\right)y\text{\u2033}+x{y}^{\prime}-3y=\text{cos}\phantom{\rule{0.1em}{0ex}}x$
$y\text{\u2033}+\frac{4}{x}{y}^{\prime}-8xy=5{x}^{2}+1$
$y\text{\u2033}+\left(\text{sin}\phantom{\rule{0.1em}{0ex}}x\right){y}^{\prime}-xy=4y$
linear, homogeneous
$y\text{\u2033}+\left(\frac{x+3}{y}\right){y}^{\prime}=0$
For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of c _{1} and c _{2} . What do the solutions have in common?
[T] $y\text{\u2033}+2{y}^{\prime}-3y=0;$ $y(x)={c}_{1}{e}^{x}+{c}_{2}{e}^{\mathrm{-3}x}$
[T] ${x}^{2}y\text{\u2033}-2y-3{x}^{2}+1=0;$ $y(x)={c}_{1}{x}^{2}+{c}_{2}{x}^{\mathrm{-1}}+{x}^{2}\text{ln}\phantom{\rule{0.1em}{0ex}}(x)+\frac{1}{2}$
[T] $y\text{\u2033}+14{y}^{\prime}+49y=0;$ $y(x)={c}_{1}{e}^{\mathrm{-7}x}+{c}_{2}x{e}^{\mathrm{-7}x}$
[T] $6y\text{\u2033}-49{y}^{\prime}+8y=0;$ $y(x)={c}_{1}{e}^{x\text{/}6}+{c}_{2}{e}^{8x}$
Find the general solution to the linear differential equation.
$y\text{\u2033}-3{y}^{\prime}-10y=0$
$y={c}_{1}{e}^{5x}+{c}_{2}{e}^{\mathrm{-2}x}$
$y\text{\u2033}-7{y}^{\prime}+12y=0$
$y\text{\u2033}+4{y}^{\prime}+4y=0$
$y={c}_{1}{e}^{\mathrm{-2}x}+{c}_{2}x{e}^{\mathrm{-2}x}$
$4y\text{\u2033}-12{y}^{\prime}+9y=0$
$2y\text{\u2033}-3{y}^{\prime}-5y=0$
$y={c}_{1}{e}^{5x\text{/}2}+{c}_{2}{e}^{\text{\u2212}x}$
$3y\text{\u2033}-14{y}^{\prime}+8y=0$
$y\text{\u2033}+{y}^{\prime}+y=0$
$y={e}^{\text{\u2212}x\text{/}2}\left({c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}\frac{\sqrt{3}x}{2}+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}\frac{\sqrt{3}x}{2}\right)$
$5y\text{\u2033}+2{y}^{\prime}+4y=0$
$y\text{\u2033}-121y=0$
$y={c}_{1}{e}^{\mathrm{-11}x}+{c}_{2}{e}^{11x}$
$8y\text{\u2033}+14{y}^{\prime}-15y=0$
$y\text{\u2033}+81y=0$
$y={c}_{1}\text{cos}\phantom{\rule{0.1em}{0ex}}9x+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}9x$
$y\text{\u2033}-{y}^{\prime}+11y=0$
$y\text{\u2033}-6{y}^{\prime}+9y=0$
$3y\text{\u2033}-2{y}^{\prime}-7y=0$
$y={c}_{1}{e}^{\left(\left(1+\sqrt{22}\right)\text{/}3\right)x}+{c}_{2}{e}^{\left(\left(1-\sqrt{22}\right)\text{/}3\right)x}$
$4y\text{\u2033}-10{y}^{\prime}=0$
$36\frac{{d}^{2}y}{d{x}^{2}}+12\frac{dy}{dx}+y=0$
$y={c}_{1}{e}^{\text{\u2212}x\text{/}6}+{c}_{2}x{e}^{\text{\u2212}x\text{/}6}$
$25\frac{{d}^{2}y}{d{x}^{2}}-80\frac{dy}{dx}+64y=0$
$\frac{{d}^{2}y}{d{x}^{2}}-9\frac{dy}{dx}=0$
$y={c}_{1}+{c}_{2}{e}^{9x}$
$4\frac{{d}^{2}y}{d{x}^{2}}+8y=0$
Solve the initial-value problem.
$y\text{\u2033}+5{y}^{\prime}+6y=0,\phantom{\rule{2em}{0ex}}y(0)=0,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{y}^{\prime}(0)=\mathrm{-2}$
$y=\mathrm{-2}{e}^{\mathrm{-2}x}+2{e}^{\mathrm{-3}x}$
$y\text{\u2033}+2{y}^{\prime}-8y=0,\phantom{\rule{2em}{0ex}}y(0)=5,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{y}^{\prime}(0)=4$
$y\text{\u2033}+4y=0,\phantom{\rule{4em}{0ex}}y(0)=3,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{y}^{\prime}(0)=10$
$y=3\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(2x\right)+5\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(2x\right)$
$y\text{\u2033}-18{y}^{\prime}+81y=0,\phantom{\rule{2em}{0ex}}y(0)=1,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{y}^{\prime}(0)=5$
$y\text{\u2033}-{y}^{\prime}-30y=0,\phantom{\rule{2em}{0ex}}y(0)=1,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{y}^{\prime}(0)=\mathrm{-16}$
$y=\text{\u2212}{e}^{6x}+2{e}^{\mathrm{-5}x}$
$4y\text{\u2033}+4{y}^{\prime}-8y=0,\phantom{\rule{2em}{0ex}}y(0)=2,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{y}^{\prime}(0)=1$
$25y\text{\u2033}+10{y}^{\prime}+y=0,\phantom{\rule{2em}{0ex}}y(0)=2,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{y}^{\prime}(0)=1$
$y=2{e}^{\text{\u2212}x\text{/}5}+\frac{7}{5}x{e}^{\text{\u2212}x\text{/}5}$
$y\text{\u2033}+y=0,\phantom{\rule{4em}{0ex}}y(\pi )=1,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}{y}^{\prime}(\pi )=\mathrm{-5}$
Solve the boundary-value problem, if possible.
$y\text{\u2033}+{y}^{\prime}-42y=0,\phantom{\rule{2em}{0ex}}y(0)=0,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}y(1)=2$
$y=\left(\frac{2}{{e}^{6}-{e}^{\mathrm{-7}}}\right){e}^{6x}-\left(\frac{2}{{e}^{6}-{e}^{\mathrm{-7}}}\right){e}^{\mathrm{-7}x}$
$9y\text{\u2033}+y=0,\phantom{\rule{4em}{0ex}}y(\frac{3\pi}{2})=6,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}y(0)=\mathrm{-8}$
$y\text{\u2033}+10{y}^{\prime}+34y=0,\phantom{\rule{2em}{0ex}}y(0)=6,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}y(\pi )=2$
No solutions exist.
$y\text{\u2033}+7{y}^{\prime}-60y=0,\phantom{\rule{2em}{0ex}}y(0)=4,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}y(2)=0$
$y\text{\u2033}-4{y}^{\prime}+4y=0,\phantom{\rule{2em}{0ex}}y(0)=2,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}y(1)=\mathrm{-1}$
$y=2{e}^{2x}-\frac{2{e}^{2}+1}{{e}^{2}}x{e}^{2x}$
$y\text{\u2033}-5{y}^{\prime}=0,\phantom{\rule{2em}{0ex}}y(0)=3,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}y(\mathrm{-1})=2$
$y\text{\u2033}+9y=0,\phantom{\rule{2em}{0ex}}y(0)=4,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}y\left(\frac{\pi}{3}\right)=\mathrm{-4}$
$y=4\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}3x+{c}_{2}\text{sin}\phantom{\rule{0.1em}{0ex}}3x,\phantom{\rule{0.2em}{0ex}}\text{infinitely many solutions}$
$4y\text{\u2033}+25y=0,\phantom{\rule{2em}{0ex}}y(0)=2,\phantom{\rule{0.2em}{0ex}}\text{}\phantom{\rule{0.2em}{0ex}}y(2\pi )=\mathrm{-2}$
Find a differential equation with a general solution that is $y={c}_{1}{e}^{x\text{/}5}+{c}_{2}{e}^{\mathrm{-4}x}.$
$5y\text{\u2033}+19{y}^{\prime}-4y=0$
Find a differential equation with a general solution that is $y={c}_{1}{e}^{x}+{c}_{2}{e}^{\mathrm{-4}x\text{/}3}.$
For each of the following differential equations:
$y\text{\u2033}+64y=0;\phantom{\rule{2em}{0ex}}y(0)=3,\phantom{\rule{1em}{0ex}}{y}^{\prime}(0)=16$
a.
$y=3\phantom{\rule{0.1em}{0ex}}\text{cos}(8x)+2\phantom{\rule{0.1em}{0ex}}\text{sin}(8x)$
b.
$y\text{\u2033}-2{y}^{\prime}+10y=0\phantom{\rule{2em}{0ex}}y(0)=1,\phantom{\rule{1em}{0ex}}{y}^{\prime}(0)=13$
$y\text{\u2033}+5{y}^{\prime}+15y=0\phantom{\rule{2em}{0ex}}y(0)=\mathrm{-2},\phantom{\rule{1em}{0ex}}{y}^{\prime}(0)=7$
a.
$y={e}^{(\mathrm{-5}\text{/}2)x}\left[\mathrm{-2}\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(\frac{\sqrt{35}}{2}x\right)+\frac{4\sqrt{35}}{35}\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(\frac{\sqrt{35}}{2}x\right)\right]$
b.
(Principle of superposition) Prove that if ${y}_{1}(x)$ and ${y}_{2}(x)$ are solutions to a linear homogeneous differential equation, $y\text{\u2033}+p(x){y}^{\prime}+q(x)y=0,$ then the function $y(x)={c}_{1}{y}_{1}(x)+{c}_{2}{y}_{2}(x),$ where ${c}_{1}$ 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 $ay\text{\u2033}+b{y}^{\prime}+cy=0$ approach zero as $x\to \infty \text{.}$ ( Hint: Consider three cases: two distinct roots, repeated real roots, and complex conjugate roots.)
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