Higher-Order Differential Equations - Differential Equations

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Question

Solve the initial value problem for and .

Answer

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^2 - 4\lambda - 5 = 0$ We then solve the characteristic equation and find that $(\lambda - 5)(\lambda + 1) = 0 ; \lambda = 5, -1$ This lets us know that the basis for the fundamental set of solutions to this problem (solutions to the homogeneous problem) contains $e^{5t}, e^{-t}$ .

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{5t} + c_2e^{-t}$ . Plugging in our initial condition, we find that . To plug in the second initial condition, we take the derivative and find that $y' = 5c_1e^{5t} - c_2e^{-t}$ . Plugging in the second initial condition yields . Solving this simple system of linear equations shows us that

Leaving us with a final answer of $y = e^{5t} - e^{-t}$

(Note, it would have been very simple to find the right answer just by taking derivatives and plugging in, but this is not overly helpful for non-multiple choice questions)

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Question

Solve the following homogeneous differential equation:

Answer

This differential equation has characteristic equation of:

It must be noted that this characteristic equation has a *double root**of r=5.*

Thus the general solution to a homogeneous differential equation with a repeated root is used.

This is equation is

$y(t)= c_1e^{r_1t}+c_2te^{r_2t}$ in the case of a repeated root such as this, and is the repeated root r=5.

Therefore, the solution is

$y(t)= c_1e^{5t}+c_2te^{5t}$

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Question

Solve the initial value problem for and

Answer

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^2 - 4\lambda + 8 = 0$ We then solve the characteristic equation and find that $(\lambda - 2)^2 + 4 = 0; \lambda = 2 \pm 2i$ (Use the quadratic formula if you'd like) This lets us know that the basis for the fundamental set of solutions to this problem (solutions to the homogeneous problem) contains $e^{2t}\sin(2t), e^{2t}\cos(2t)$ .

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{2t}\sin(2t) + c_2e^{2t}\cos(2t)$ . Plugging in our initial condition, we find that . To plug in the second initial condition, we take the derivative and find that $y' = 2c_1e^{2t}(\sin(2t) + \cos(2t)) + 2e^{2t}(\cos(2t) - sin(2t))$ . Plugging in the second initial condition yields . Solving this simple system of linear equations shows us that

Leaving us with a final answer of $y = e^{2t}\cos(2t)$

(Note, it would have been very simple to find the right answer just by taking derivatives and plugging in, but this is not overly helpful for non-multiple choice questions)

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Question

Find the general solution to .

Answer

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^3 - \lambda^2 + \lambda - 1 = 0$ To factor this, in this case we may use factoring by grouping. More generally, we may use horner's scheme/synthetic division to test possible roots. Here are both methods shown.

$\lambda^3 - \lambda^2 + \lambda - 1 = (\lambda^3 + \lambda) - (\lambda^2 + 1) = \lambda(\lambda^2 + 1) - (\lambda^2 +1) = (\lambda^2 + 1)(\lambda - 1)$

Alternatively, the rational root theorem suggests that we try -1 or 1 as a root of this equation. Using horner's scheme, we see

$1 \left |\begin{matrix}1 && -1 && 1 && -1\ && 1 && 0 && 1 \ \hline \end{matrix}\right.$

Which tells us the the polynomial factors into $(\lambda - 1)(\lambda^2 + 1)$ and that $\lambda = 1, \pm i$ . This means that the fundamental set of solutions is

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{t} + c_2\sin(t) + c_3\cos(t)$ . As this is not an initial value problem and just asks for the general solution, we are done.

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Question

Solve the following homogeneous differential equation:

Answer

The ode has a characteristic equation of .

This yields the double root of r=2. Then the roots are plugged into the general solution to a homogeneous differential equation with a repeated root.

$y= c_1e^{rt}+c_2te^{rt}$ (for real repeated roots)

Thus, the solution is,

$y(t) = c_{1}e^{2t}+c_2te^{2t}$

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Question

Solve the General form of the differential equation:

Answer

This differential equation has a characteristic equation of

, which yields the roots for r=2 and r=3. Once the roots or established to be real and non-repeated, the general solution for homogeneous linear ODEs is used. this equation is given as:

$y(t)= c_1e^{r_1t}+c_2e^{r_2t}$

with r being the roots of the characteristic equation.

Thus, the solution is

$y(t)= c_1e^{2t}+c_2e^{3t}$

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Question

Solve the general homogeneous part of the following differential equation:

Answer

We start off by noting that the homogeneous equation we are trying to solve is given as

.

This differential equation thus has characteristic equation of

.

This has roots of r=3 and r=-4, therefore, the general homogeneous solution is given by:

$y(t)= c_1e^{3t}+c_2e^{-4t}$

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Question

Find a general solution to the following Differential Equation

$y^{'''} + 2y^{''} - 11y' - 12y = 0$

Answer

Solving the auxiliary equation

Trying out candidates for roots from the Rational Root Theorem we have a root .

Factoring completely we have

Our general solution is

$y(t) = C_1e^{3x} + C_2e^{-4x} + C_3e^{-x}$

where are arbitrary constants.

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Question

For the initial value problem

$x(x-1)y^{'''} -3xy^{''} + 6x^2y' - \cos{(x)}y = \sqrt{x+5}$

Which if the following intervals containing do NOT guarantees the existence of a unique solution?

Answer

Putting the equation in standard form we get that

$y^{'''} - \frac{3}{(x-1)}y^{''} + \frac{6x}{(x-1)}y' - \frac{\cos{x}}{x(x-1)}y = \frac{\sqrt{x+5}}{x(x-1)}$

We need to find where these coefficients are simultaneously continuous. This is where

$(-5,0), (0,1), (1,\infty)$ . The choice that is not a subset of these is

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Question

For the initial value problem

$x(x-1)y^{'''} -3xy^{''} + 6x^2y' - \cos{(x)}y = \sqrt{x+5}$

Which if the following intervals containing do NOT guarantees the existence of a unique solution?

Answer

Putting the equation in standard form we get that

$y^{'''} - \frac{3}{(x-1)}y^{''} + \frac{6x}{(x-1)}y' - \frac{\cos{x}}{x(x-1)}y = \frac{\sqrt{x+5}}{x(x-1)}$

We need to find where these coefficients are simultaneously continuous. This is where

$(-5,0), (0,1), (1,\infty)$ . The choice that is not a subset of these is

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Question

Solve the initial value problem for and .

Answer

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^2 - 4\lambda - 5 = 0$ We then solve the characteristic equation and find that $(\lambda - 5)(\lambda + 1) = 0 ; \lambda = 5, -1$ This lets us know that the basis for the fundamental set of solutions to this problem (solutions to the homogeneous problem) contains $e^{5t}, e^{-t}$ .

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{5t} + c_2e^{-t}$ . Plugging in our initial condition, we find that . To plug in the second initial condition, we take the derivative and find that $y' = 5c_1e^{5t} - c_2e^{-t}$ . Plugging in the second initial condition yields . Solving this simple system of linear equations shows us that

Leaving us with a final answer of $y = e^{5t} - e^{-t}$

(Note, it would have been very simple to find the right answer just by taking derivatives and plugging in, but this is not overly helpful for non-multiple choice questions)

Compare your answer with the correct one above

Question

Solve the following homogeneous differential equation:

Answer

This differential equation has characteristic equation of:

It must be noted that this characteristic equation has a *double root**of r=5.*

Thus the general solution to a homogeneous differential equation with a repeated root is used.

This is equation is

$y(t)= c_1e^{r_1t}+c_2te^{r_2t}$ in the case of a repeated root such as this, and is the repeated root r=5.

Therefore, the solution is

$y(t)= c_1e^{5t}+c_2te^{5t}$

Compare your answer with the correct one above

Question

Solve the initial value problem for and

Answer

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^2 - 4\lambda + 8 = 0$ We then solve the characteristic equation and find that $(\lambda - 2)^2 + 4 = 0; \lambda = 2 \pm 2i$ (Use the quadratic formula if you'd like) This lets us know that the basis for the fundamental set of solutions to this problem (solutions to the homogeneous problem) contains $e^{2t}\sin(2t), e^{2t}\cos(2t)$ .

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{2t}\sin(2t) + c_2e^{2t}\cos(2t)$ . Plugging in our initial condition, we find that . To plug in the second initial condition, we take the derivative and find that $y' = 2c_1e^{2t}(\sin(2t) + \cos(2t)) + 2e^{2t}(\cos(2t) - sin(2t))$ . Plugging in the second initial condition yields . Solving this simple system of linear equations shows us that

Leaving us with a final answer of $y = e^{2t}\cos(2t)$

(Note, it would have been very simple to find the right answer just by taking derivatives and plugging in, but this is not overly helpful for non-multiple choice questions)

Compare your answer with the correct one above

Question

Find the general solution to .

Answer

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

$\lambda^3 - \lambda^2 + \lambda - 1 = 0$ To factor this, in this case we may use factoring by grouping. More generally, we may use horner's scheme/synthetic division to test possible roots. Here are both methods shown.

$\lambda^3 - \lambda^2 + \lambda - 1 = (\lambda^3 + \lambda) - (\lambda^2 + 1) = \lambda(\lambda^2 + 1) - (\lambda^2 +1) = (\lambda^2 + 1)(\lambda - 1)$

Alternatively, the rational root theorem suggests that we try -1 or 1 as a root of this equation. Using horner's scheme, we see

$1 \left |\begin{matrix}1 && -1 && 1 && -1\ && 1 && 0 && 1 \ \hline \end{matrix}\right.$

Which tells us the the polynomial factors into $(\lambda - 1)(\lambda^2 + 1)$ and that $\lambda = 1, \pm i$ . This means that the fundamental set of solutions is

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, $y = c_1e^{t} + c_2\sin(t) + c_3\cos(t)$ . As this is not an initial value problem and just asks for the general solution, we are done.

Compare your answer with the correct one above

Question

Solve the following homogeneous differential equation:

Answer

The ode has a characteristic equation of .

This yields the double root of r=2. Then the roots are plugged into the general solution to a homogeneous differential equation with a repeated root.

$y= c_1e^{rt}+c_2te^{rt}$ (for real repeated roots)

Thus, the solution is,

$y(t) = c_{1}e^{2t}+c_2te^{2t}$

Compare your answer with the correct one above

Question

Solve the General form of the differential equation:

Answer

This differential equation has a characteristic equation of

, which yields the roots for r=2 and r=3. Once the roots or established to be real and non-repeated, the general solution for homogeneous linear ODEs is used. this equation is given as:

$y(t)= c_1e^{r_1t}+c_2e^{r_2t}$

with r being the roots of the characteristic equation.

Thus, the solution is

$y(t)= c_1e^{2t}+c_2e^{3t}$

Compare your answer with the correct one above

Question

Solve the general homogeneous part of the following differential equation:

Answer

We start off by noting that the homogeneous equation we are trying to solve is given as

.

This differential equation thus has characteristic equation of

.

This has roots of r=3 and r=-4, therefore, the general homogeneous solution is given by:

$y(t)= c_1e^{3t}+c_2e^{-4t}$

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Question

Find a general solution to the following Differential Equation

$y^{'''} + 2y^{''} - 11y' - 12y = 0$

Answer

Solving the auxiliary equation

Trying out candidates for roots from the Rational Root Theorem we have a root .

Factoring completely we have

Our general solution is

$y(t) = C_1e^{3x} + C_2e^{-4x} + C_3e^{-x}$

where are arbitrary constants.

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Question

Solve the given differential equation by undetermined coefficients.

Answer

First solve the homogeneous portion:

$\y''-8y'+16y=0 \m^2-8m+16=0 \(m-4)(m-4)=0$

Therefore, is a repeated root thus one of the complimentary solutions is,

$y_c=c_1e^{4x}+c_2xe^{4x}$

Now find the remaining complimentary solution .

$\y_p=Ax+B \y_p'=A \y_p''=0$

Now solve for and .

$\0-8A+16(Ax+B)=24x+2 \-8A+16Ax+16B=24x+2$

Where

$\16A=24 \\A=\frac{24}{16}=\frac{3}{2}$

and

$\-8A+16B=2 \\-8\left(\frac{3}{2} \right )+16B=2 \\-\frac{24}{2}+16B=2 \\-12+16B=2 \\16B=14 \\B=\frac{14}{16}=\frac{7}{8}$

Therefore,

$y_p=\frac{3}{2}x+\frac{7}{8}$

Now, combine both of the complimentary solutions together to arrive at the general solution.

$\y=y_c+y_p \y=c_1e^{4x}+c_2xe^{4x}+\frac{3}{2}x+\frac{7}{8}$

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Question

Find the general solution for $y'' - 3y' - 4y = -25\cos(2t)$ .

Answer

This is a higher order inhomogeneous linear differential equation. Because the inhomogeneity is a cosine, we will use variation of parameters to solve it.

First, we find the characteristic equation to solve for the homogenous solution. This gives us $\lambda^2 - 3\lambda - 4 = (\lambda -4)(\lambda + 1) = 0$ .

This tells us that the homogeneous solution is $c_1e^{4t} + c_2e^{-t}$ . As neither of these overlap with our inhomogeneity, we are safe to continue without adding a factor of t.

Thus, let us guess that $y_p = a_1\sin(2t) + a_2\cos(2t)$ . Then,

$y_p' = 4a_1\cos(2t) - 4a_2\sin(2t)$ and

$y_p'' = -4a_1\sin(2t) - 4a_2\cos(2t)$

Plugging into the original equation, we have

$\sin(2t)(-4a_1 + 6a_2 -4a_1) + \cos(2t)(-4a_2 - 6a_1 -4a_2) = -25\cos(2t)$

Which implies that and . Solving via substitution,

$a_1 = \frac{3}{4}a_2$

$-8a_2 - \frac{9}{2}a_2 = -\frac{25}{2}a_2 = -25; a_2 = 2$

$a_1 = \frac{3}{2}$

Thus, the particular solution is $y_p = \frac{3}{2}\sin(2t) + 2\cos(2t)$ and the overall solution is the particular plus the homogeneous.

So

$y = c_1e^{4t} + c_2e^{-t} + \frac{3}{2}\sin(3t) + 2\cos(3t)$

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