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Problem 27

Generalized Blasius Equation. H. Blasius, in his study of

laminar flow of a fluid, encountered an equation of the form

$$

y^{\prime \prime \prime}+y y^{\prime \prime}=\left(y^{\prime}\right)^{2}-1

$$

Use the Runge-Kutta algorithm for systems with $h=0.1$

to approximate the solution that satisfies the initial con-

ditions $y(0)=0, y^{\prime}(0)=0,$ and $y^{\prime \prime}(0)=1.32824$

Sketch this solution on the interval $[0,2]$ .

Answer

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## Recommended Questions

Using the vectorized Runge-Kutta algorithm for systems with $h=0.125$ approximate the solution to the initial value problem

$$\begin{array}{ll}{x^{\prime}=2 x-y ;} & {x(0)=0} \\ {y^{\prime}=3 x+6 y ;} & {y(0)=-2}\end{array}$$

at $t=1$ Compare this approximation to the actual solution

$$x(t)=e^{5 t}-e^{3 t}, \quad y(t)=e^{3 t}-3 e^{5 t}$$

Use the fourth-order Runge-Kutta subroutine with h = 0.1 to approximate the solution to

$$ y^{\prime}=3 \cos (y-5 x), \quad y(0)=0 $$

at the points x = 0, 0.1, 0.2, . . . , 4.0. Use your answers to make a rough sketch of the solution on [0, 4].

Find the general solution to the given differential equation on the interval $(0, \infty).$

$$x^{2} y^{\prime \prime}-x y^{\prime}=0.$$

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.

$y(x)=e^{a x}\left(c_{1}+c_{2} x\right), \quad y^{\prime \prime}-2 a y^{\prime}+a^{2} y=0,$ where $a$ is a constant.

Find the general solution to the given differential equation on the interval $(0, \infty).$

$$x^{2} y^{\prime \prime}+9 x y^{\prime}+16 y=0.$$

Find the unique solution satisfying the differential equation and the initial conditions given, where $y_{p}(x)$ is the particular solution.

$y^{\prime \prime}-2 y^{\prime}+y=12 e^{x}$, $\mathrm{y}_{p}(x)=6 x^{2} e^{x}$, $y(0)=-1, y^{\prime}(0)=0$

Find the general solution to the given differential equation on the interval $(0, \infty).$

$$x^{2} y^{\prime \prime}+9 x y^{\prime}+15 y=0.$$

The Korteweg-deVries equation This nonlinear differential equation, which describes wave motion on shallow water surfaces, is given by

$$4 u_{t}+u_{x x x}+12 u u_{x}=0.$$

Show that $u(x, t)=\operatorname{sech}^{2}(x-t)$ satisfies the Kortweg-deVries equation.

Find the particular solution to the differential equation $y^{\prime}\left(1-x^{2}\right)=1+y$ that passes through $(0,-2),$ given that $y=C \frac{\sqrt{x+1}}{\sqrt{1-x}}-1$ is a general solution.

Use the variation-of-parameters method to find the general solution to the given differential equation.

$$y^{\prime \prime}+2 y^{\prime}+y=\frac{e^{-x}}{\sqrt{4-x^{2}}}, \quad|x|<2$$

Determine a particular solution to the given differential equation of the form

$$

y_{p}(x)=A_{0}+A_{1} x+A_{2} x^{2}

$$

Also find the general solution to the differential equation:

$$

y^{\prime \prime}+y^{\prime}-2 y=4 x^{2}+5

$$

Find the general solution to the given differential equation on the interval $(0, \infty).$

$$x^{2} y^{\prime \prime}+x y^{\prime}+25 y=0.$$

Let $L[y] :=y^{\prime \prime \prime}-x y^{\prime \prime}+4 y^{\prime}-3 x y, y_{1}(x)=\cos 2 x,$ and

$y_{2}(x) :=-1 / 3 .$ Verify that $L\left[y_{1}\right](x)=x \cos 2 x$ and

$L\left[y_{2}\right](x)=x .$ Then use the superposition principle (linearity) to find a solution to the differential equation:

(a) $L[y]=7 x \cos 2 x-3 x$ .

(b) $L[y]=-6 x \cos 2 x+11 x$ .

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.

$$y(x)=c_{1} x^{-3}+c_{2} x^{-1}, \quad x^{2} y^{\prime \prime}+5 x y^{\prime}+3 y=0$$.

Solve the initial value problems for $\mathbf{r}$ as a vector function of $t.$

Differential equation: $\quad \frac{d^{2} \mathbf{r}}{d t^{2}}=e^{i} \mathbf{i}-e^{-t} \mathbf{j}+4 e^{2 t} \mathbf{k}$

Initial conditions: $\quad \mathbf{r}(0)=3 \mathbf{i}+\mathbf{j}+2 \mathbf{k}$ and

$\left.\frac{d \mathbf{r}}{d t}\right|_{t=0}=-\mathbf{i}+4 \mathbf{j}$

For the following problems, use the direction field below from the differential equation $y^{\prime}=y^{2}-2 y .$ Sketch the graph of the solution for the given initial conditions.

$$y(0)=1$$

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.

$$y(x)=c_{1} \cos 2 x+c_{2} \sin 2 x, \quad y^{\prime \prime}+4 y=0$$.

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.

$$y(x)=c_{1} e^{x}+c_{2} e^{-2 x}, \quad y^{\prime \prime}+y^{\prime}-2 y=0$$.

Solve the initial value problems for $\mathbf{r}$ as a vector function of $t.$

Differential equation: $\quad \frac{d^{2} \mathbf{r}}{d t^{2}}=-32 \mathbf{k}$

Initial conditions: $\quad \mathbf{r}(0)=100 \mathbf{k}$ and$$\left.\frac{d \mathbf{r}}{d t}\right|_{t=0}=8 \mathbf{i}+8 \mathbf{j}$$

Find the general solution to the linear differential equation.

$$

2 y^{\prime \prime}=0

$$

For all problems below, use a complex-valued trial solution to determine a particular solution to the given differential equation.

$$y^{\prime \prime}-y^{\prime}-2 y=40 \sin ^{2} x$$

Solve the initial value problems for $\mathbf{r}$ as a vector function of $t.$

Differential equation: $\quad \frac{d^{2} \mathbf{r}}{d t^{2}}=-(\mathbf{i}+\mathbf{j}+\mathbf{k})$

Initial conditions: $\quad \mathbf{r}(0)=10 \mathbf{i}+10 \mathbf{j}+10 \mathbf{k}$ and

$\left.\frac{d \mathbf{r}}{d t}\right|_{t=0}=0$

Fluid Flow. The streamlines associated with a cer-

tain fluid flow are represented by the family of curves $$y=x-1+k e^{-x}$$. The velocity potentials of the flow are just the orthogonal trajectories of this family.

(a) Use the method described in Problem 32 of Exer-

cises 2.4 to show that the velocity potentials satisfy

$$d x+(x-y) d y=0$$

[Hint: First express the family $$y=x-1+k e^{-x}$$ in the form $$F(x, y)=k . ]$$

(b) Find the velocity potentials by solving the equation

obtained in part (a).

Use the variation-of-parameters method to find the general solution to the given differential equation.

$$y^{\prime \prime}+4 y^{\prime}+4 y=\frac{4 e^{-2 x}}{1+x^{2}}+2 x^{2}-1$$

Find the general solution to the given differential equation on the interval $(0, \infty).$

$$x^{2} y^{\prime \prime}-11 x y^{\prime}+37 y=0.$$

Use the variation-of-parameters technique to find a particular solution $\mathbf{x}_{p}$ to $\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b},$ for the given $A$ and $\mathbf{b} .$ Also obtain the general solution to the system of differential equations.

$$A=\left[\begin{array}{rr}

2 & -1 \\

-1 & 2

\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c}

0 \\

4 e^{t}

\end{array}\right]$$

Use the variation-of-parameters method to find the general solution to the given differential equation.

$$y^{\prime \prime}+y=\csc x+2 x^{2}+5 x+1, \quad 0<x<\pi$$

Determine the general solution to the given differential equation on $(0, \infty)$

$$x^{2} y^{\prime \prime}-x y^{\prime}+5 y=0$$

The following equation can be used to relate the density of liquid water to Celsius temperature in the range from $0^{\circ} \mathrm{C}$ to about $20^{\circ} \mathrm{C}:$

$$d\left(\mathrm{g} / \mathrm{cm}^{3}\right)=\frac{0.99984+\left(1.6945 \times 10^{-2} t\right)-\left(7.987 \times 10^{-6} t^{2}\right)}{1+\left(1.6880 \times 10^{-2} t\right)}$$

(a) To four significant figures, determine the density of water at $10^{\circ} \mathrm{C}$

(b) At what temperature does water have a density of $0.99860 \mathrm{g} / \mathrm{cm}^{3} ?$

(c) In the following ways, show that the density passes through a maximum somewhere in the temperature range to which the equation applies.

(i) by estimation

(ii) by a graphical method

(iii) by a method based on differential calculus.

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.

$$y(x)=c_{1} x^{1 / 2}+3 x^{2}, \quad 2 x^{2} y^{\prime \prime}-x y^{\prime}+y=9 x^{2}$$.

Use the variation-of-parameters method to find the general solution to the given differential equation.

$$y^{\prime \prime}+4 y^{\prime}+4 y=x^{-2} e^{-2 x}, \quad x>0$$

Determine the general solution to the given differential equation on $(0, \infty)$

$$x^{2} y^{\prime \prime}-x y^{\prime}-35 y=0$$

Sketch several solution curves in the phase plane of the system of differential equations $d \mathbf{x} / d t=A \mathbf{x}$ using the given eigenvalues and eigenvectors of $A .$

$\lambda_{1}=-1, \quad \lambda_{2}=-2 ; \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{r}{-1} \\ {1}\end{array}\right]$

Determine the general solution to the given differential equation on $(0, \infty)$

$$x^{2} y^{\prime \prime}+5 x y^{\prime}+13 y=0$$

Solve the given differential equation on the interval $x>0 .$ [Remember to put the equation in standard form.]

$$x^{2} y^{\prime \prime}+4 x y^{\prime}+2 y=\cos x$$

Determine a particular solution to the given differential equation of the form $y_{p}(x)=A_{0} e^{2 x} .$ Also find the general solution to the differential equation:

$$

y^{\prime \prime \prime}+2 y^{\prime \prime}-y^{\prime}-2 y=4 e^{2 x}

$$

Verify that the given function is a solution to the given differential equation. In these problems, $c_{1}$ and $c_{2}$ are arbitrary constants.

$$\begin{aligned}

&\diamond y(x)=c_{1} e^{x}+c_{2} e^{-x}\left(1+2 x+2 x^{2}\right)\\

&x y^{\prime \prime}-2 y^{\prime}+(2-x) y=0, x>0

\end{aligned}$$.

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.

$y(x)=e^{a x}\left(c_{1} \cos b x+c_{2} \sin b x\right)$

$y^{\prime \prime}-2 a y^{\prime}+\left(a^{2}+b^{2}\right) y=0,$ where $a$ and $b$ are

constants.

Use the strategy of Example 3 to find a value of $h$ for Euler's method such that $y(1)$ is approximated to within$\pm 0.01,$ if $y(x)$ satisfies the initial value problem

$$y^{\prime}=x-y, \quad y(0)=0$$

Also find, to within $\pm 0.05,$ the value of $x_{0}$ such that $y\left(x_{0}\right)=0.2 .$ Compare your answers with those given by the actual solution $y=e^{-x}+x-1$ (verify!). Graph the polygonal-line approximation and the actual solution on the same coordinate system.

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

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.

$$y(x)=c_{1} x^{2} \ln x, \quad x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0$$.

The Bessel equation of order one-half

$$t^{2} y^{\prime \prime}+t y^{\prime}+\left(t^{2}-\frac{1}{4}\right) y=0, \quad t>0$$

has two linearly independent solutions, $y_{1}(t)=t^{-1 / 2} \cos t, \quad y_{2}(t)=t^{-1 / 2} \sin t.$

Find a general solution to the nonhomogeneous equation $t^{2} y^{\prime \prime}+t y^{\prime}+\left(t^{2}-\frac{1}{4}\right) y=t^{5 / 2}, \quad t>0.$

Solve the given differential equation on the interval $x>0 .$ [Remember to put the equation in standard form.]

$$x^{2} y^{\prime \prime}+6 x y^{\prime}+6 y=4 e^{2 x}$$

Verify that the given function is a solution to the given differential equation $\left(c_{1} \text { and } c_{2}\right.$ are arbitrary constants), and state the maximum interval over which the solution is valid.

$$y(x)=c_{1} x^{2} \cos (3 \ln x), \quad x^{2} y^{\prime \prime}-3 x y^{\prime}+13 y=0$$.