Determine whether existence of at least one solution of given initial value problem is guaranteed and is so, whether solution is unique. \frac{dy}{dx}=x\ln y;y\left ( 1\right ) =1
Solution
f\left ( x,y\right ) =x\ln y
Determine whether existence of at least one solution of given initial value problem is guaranteed and is so, whether solution is unique. \frac{dy}{dx}=x-1;y\left ( 0\right ) =1
Solution
f\left ( x,y\right ) =x-1
Determine whether existence of at least one solution of given initial value problem is guaranteed and is so, whether solution is unique. y\frac{dy}{dx}=x-1;y\left ( 1\right ) =0
Solution
f\left ( x,y\right ) =\frac{x-1}{y}
Use the method of example 2 (page 20) to construct slope field then sketch solution curve corresponding to the given initial condition. Finally use this solution curve to estimate the desired value of the solution y(x). \begin{align*} \frac{dy}{dx} & =y-x\\ y\left ( 4\right ) & =1\\ y\left ( -4\right ) & =? \end{align*}
Solution
f\left ( x,y\right ) =y-x
Suppose the deer population P\left ( t\right ) in small forest satisfies logistic equation \frac{dp}{dt}=0.0225p-0.0003p^{2}. Construct a slope field and appropriate solution curve to answer the following questions: If there are 25 deer at time t=0 and t is measured in months, how long will it take for the number of deer to double? What will be the limiting deer population?
Solution
The slope field was first drawn. Then the point \left ( 0,25\right ) was located. Then the slope field was traced until y=50, which is double the number of deer from the initial starting time. Now the t component was read from the slope field to answer the first part of the question. f\left ( t,p\right ) =0.0225p-0.0003p^{2}
Verify that if k is constant, then the function y\left ( x\right ) =kx satisfies the differential equation xy^{\prime }=y for all x. Construct a slope field and several of the these straight line solution curves. Then determine (in terms of a and b) how many different solutions the initial value problem xy^{\prime }=y;y\left ( a\right ) =b has. One, none or infinitely many.
Solution
To verify that y\left ( x\right ) =kx satisfies the differential equation, we plug-in this solution into the ODE and check that we get the same RHS as given. We see that y^{\prime }\left ( x\right ) =k. Therefore xy^{\prime }=y becomes x\left ( k\right ) =y=kx. Hence satisfied. f\left ( x,y\right ) =\frac{y}{x}
We see from the above, that if we start from x=0,y=0, then there are \infty number of solutions, since there are \infty number of slope lines starting or ending at \left ( 0,0\right ) . For any point \left ( a,b\right ) where a\neq 0, there is unique solution, since we can find interval around \left ( a,b\right ) in this case with unique slope line. Finally, if a=0\,\ but b\neq 0, which means the initial condition is at the y axis, then there is no solution, since the slop is \infty in this case. Hence
Verify that if c is constant, then the function defined piecewise by y\left ( x\right ) =\left \{ \begin{array} [c]{ccc}1 & & x\leq c\\ \cos \left ( x-c\right ) & & c<x<c+\pi \\ -1 & & x\geq c+\pi \end{array} \right .
Solution
The solution y\left ( x\right ) is plotted for c=0,-1,+1. The following show the result. The effect of c is that it causes a shift to the left or right depending on value of c.
Since f\left ( x,y\right ) =-\sqrt{1-y^{2}}
Hence satisfied. When x\geq c+\pi then y\left ( x\right ) =-1 and plugging this into the ODE gives \begin{align*} 0 & =-\sqrt{1-\left ( -1\right ) ^{2}}\\ & =-\sqrt{1-1}\\ & =0 \end{align*}
Hence solution y\left ( x\right ) satisfies the ODE. The slope field is now plotted
We see from the slope plot, that starting at any point in a region, as long as \left \vert y\right \vert <0\,, then the solution is unique. When y=1 or y=-1, then y^{\prime }=0, and this gives infinite number of solutions since y=c for any constant is a solution. For real solution, y can not be larger than 1. Hence in summary
Final general solution of \frac{dy}{dx}=3\sqrt{xy}
Solution
This is separable.\begin{align*} \frac{dy}{\sqrt{y}} & =3\sqrt{x}dx\\ y^{\frac{-1}{2}}dy & =3x^{\frac{1}{2}}dx \end{align*}
Integrating\begin{align*} \frac{y^{\frac{1}{2}}}{\frac{1}{2}} & =3\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+c\\ 2y^{\frac{1}{2}} & =2x^{\frac{3}{2}}+c\\ y^{\frac{1}{2}} & =x^{\frac{3}{2}}+c_{1}\\ y & =\left ( x^{\frac{3}{2}}+c_{1}\right ) ^{2} \end{align*}
Final general solution of \left ( 1+x\right ) ^{2}\frac{dy}{dx}=\left ( 1+y\right ) ^{2}
Solution
Before solving, it is good idea to check if the solution exist and if it is unique. f\left ( x,y\right ) =\frac{\left ( 1+y\right ) ^{2}}{\left ( 1+x\right ) ^{2}}
Now the ODE is solved. This is separable. \frac{dy}{\left ( 1+y\right ) ^{2}}=\frac{dx}{\left ( 1+x\right ) ^{2}}
Hence \fbox{$y=\frac{1+x}{1+c_1\left ( 1+x\right ) }-1$}
Find explicit particular solution of \frac{dy}{dx}=4x^{3}y-y;y\left ( 1\right ) =-3
Solution
Before solving, it is good idea to check if the solution exist and if it is unique. f\left ( x,y\right ) =4x^{3}y-y
Let e^{c}=c_{1}, then the above can be written as y=c_{1}e^{x^{4}-x}
Find explicit particular solution of \frac{dy}{dx}=2xy^{2}+3x^{2}y^{2};y\left ( 1\right ) =-1
Solution
Before solving, it is good idea to check if the solution exist and if it is unique. f\left ( x,y\right ) =2xy^{2}+3x^{2}y^{2}
Applying initial conditions to find c gives\begin{align*} -1 & =\frac{-1}{1+1+c}\\ -2-c & =-1\\ c & =-1 \end{align*}
Hence solution is\begin{align*} y & =\frac{-1}{x^{2}+x^{3}-1}\\ & =\frac{1}{1-x^{2}-x^{3}} \end{align*}
Here is a plot of the solution in small interval around x=1
We notice that at the real root of 1-x^{2}-x^{3}, the solution y\left ( x\right ) goes to \pm \infty . This happens at x\approx 0.75487.
Solve \left ( \frac{dy}{dx}\right ) ^{2}=4y to verify the general solution curves and singular solution curve that are illustrated in fig 1.4.5. Then determine the points \left ( a,b\right ) in the plane for which the initial value problem \left ( y^{\prime }\right ) ^{2}=4y;y\left ( a\right ) =b has (a) No solution, (b) infinitely many solutions that are defined for all x, (c) on some neighborhood of the point x=a, only finitely many solutions.
Solution
Figure 1.4.5 is below
f\left ( x,y\right ) =\pm 2\sqrt{y}
\frac{dy}{dx}=\pm 2\sqrt{y}
For the positive case\begin{align*} y^{\frac{-1}{2}}dy & =2dx\\ 2y^{\frac{1}{2}} & =2x+c\\ y^{\frac{1}{2}} & =x+c_{1}\\ y & =\left ( c_{1}+x\right ) ^{2} \end{align*}
Hence the solutions are y\left ( x\right ) =\left \{ \begin{array} [c]{ccc}\left ( c_{1}-x\right ) ^{2} & & \\ \left ( c_{1}+x\right ) ^{2} & & \\ 0 & & \text{singular solution}\end{array} \right .
The following is plot of y\left ( x\right ) =\left ( c_{1}-x\right ) ^{2} for few values of c_{1} to show the shape of the solution curves. This agrees with the figure given in the book.
A certain moon rock was found to contain equal number of potassium and argon atoms. Assume that all the argon is the result of radioactive decay of potassium (its half like is about 1.28\times 10^{9} years) and that one of every nine potassium atom disintegrations yields an argon atom. What is the age of the rock, measured from the time it contained only potassium?
Solution
Half life is the time for a quantity to reduce to half its original number. Let T=1.28\times 10^{9} years in this example. Let P\left ( 0\right ) be the number of potassium atoms at time t=0. Hence the formula for half life decay is P\left ( t\right ) =P\left ( 0\right ) \left ( \frac{1}{2}\right ) ^{\frac{t}{T}}
Since we want to find t when g\left ( t\right ) =P\left ( t\right ) , then we solve from\begin{align*} g\left ( t\right ) & =P\left ( t\right ) \\ \frac{1}{9}P\left ( 0\right ) \left ( 1-\left ( \frac{1}{2}\right ) ^{\frac{t}{T}}\right ) & =P\left ( 0\right ) \left ( \frac{1}{2}\right ) ^{\frac{t}{T}}\\ \frac{1}{9}\left ( 1-\left ( \frac{1}{2}\right ) ^{\frac{t}{T}}\right ) & =\left ( \frac{1}{2}\right ) ^{\frac{t}{T}}\\ 1-\left ( \frac{1}{2}\right ) ^{\frac{t}{T}} & =9\left ( \frac{1}{2}\right ) ^{\frac{t}{T}}\\ 1 & =9\left ( \frac{1}{2}\right ) ^{\frac{t}{T}}+\left ( \frac{1}{2}\right ) ^{\frac{t}{T}}\\ 1 & =10\left ( \frac{1}{2}\right ) ^{\frac{t}{T}}\\ \frac{1}{10} & =\left ( \frac{1}{2}\right ) ^{\frac{t}{T}} \end{align*}
Taking log \begin{align*} \log \left ( \frac{1}{10}\right ) & =\frac{t}{T}\log \left ( \frac{1}{2}\right ) \\ t & =T\frac{\log \left ( \frac{1}{10}\right ) }{\log \left ( \frac{1}{2}\right ) }\\ & =1.28\times 10^{9}\left ( \frac{-2.3}{-0.693}\right ) \\ & =4.248\,2\times 10^{9} \end{align*}
Hence it will take 4.2482 billion years.
The barometric pressure p (in inches of mercury) at an altitude x miles above sea level satisfies the initial value problem \frac{dp}{dx}=\left ( -0.2\right ) p;p\left ( 0\right ) =29.92. (a) Calculate the barometric pressure at 10,000 ft. and again at 30,000 ft. (b) Without prior conditioning, few people can survive when the pressure drops to less than 15 in. Of mercury. How high is that?
Solution
\frac{dp}{dx}=\left ( -0.2\right ) p
To find c, we apply initial conditions. At x=0,p=29.92 in, hence 29.92=c
when x=30000 ft or 30000/5280=5.6818 miles, then\begin{align*} p & =29.92e^{-0.2\left ( 5.6818\right ) }\\ & =9.6039\text{ in} \end{align*}
We solve for x from\begin{align*} 15 & =29.92e^{-0.2x}\\ \frac{15}{29.92} & =e^{-0.2x} \end{align*}
Taking natural log\begin{align*} \ln \frac{15}{29.92} & =-0.2x\\ -0.69047 & =-0.2x \end{align*}
Hence\begin{align*} x & =\frac{0.69047}{0.2}=3.4524\text{ miles}\\ & =\left ( 3.4524\right ) \left ( 5280\right ) =18229\text{ ft} \end{align*}