7.3.1Can I solve a diff-EQ?

Solving Differential Equations

7-96.

If $\frac{dy}{dx}=x$ , then what does $y$ equal? Is there more than one possible solution? Show how you got your answer.

7-97.

Write general solutions for $y$ .

$\frac{dy}{dx}=6\cos(x)$

$\frac { d y } { d x }= x^3$

$\frac { d y } { d x }= y$

Explain why part (c) does not follow the same pattern as parts (a) and (b) for the general solutions.

Solving Differential Equations

Any equation that contains derivatives (such as those in problem 7-97) is called a differential equation. We have seen many of these before and have usually solved them by computing a general antiderivative.

However, there is another way to solve some differential equations: by separating the differentials ( $dx$ and $dy$ ) and then integrating.

$\left. \begin{array}{l}{ \frac { d y } { d x } = x }\\{ d y = x \cdot d x }\\{ \int d y = \int x d x }\\{ y + C _ { 1 } = \frac { x ^ { 2 } } { 2 } + C _ { 2 } }\\{ y = \frac { x ^ { 2 } } { 2 } + C }\end{array} \right.$

The solution is a function that will make the original differential equation true. That is, if you differentiate the solution and substitute it for $\frac { d y } { d x }$ , you get a true statement.

7-98.

YONG LI RETURNS AGAIN!

After speaking with her financial advisor, Yong Li decides to put her money into a new bank account.

Describe what the rate of growth in her account depends on.
If her initial deposit is $B = $1000$ and the bank offers $10\%$ interest, what is the value of $\frac { d B } { d t }$ ?
Calculate $\frac { d B } { d t }$ if her balance is $B = $5600$ and the bank offers $7.5\%$ interest.
Write a differential equation relating $\frac { d B } { d t }$ to balance $B$ and interest $r=15\%$ .

7-99.

SOLVING DIFFERENTIAL EQUATIONS WITH IMPLICIT INTEGRATION

Back in Chapter 6, exponential equations were used to represent Yong Li’s bank account balance at time $t$ . However, our equation from part (d) of problem 7-98 does not look exponential! What is going on?

Let’s examine how a differential equation (such as $\frac { d B } { d t }= 0.15B$ ) relates to an exponential function. This can by done by applying the implicit integration technique from the Math Notes box to the differential equation. Follow the given solution of this differential equation as you complete the parts below.

In order to solve $\frac { d B } { d t }= 0.15B$ , separate the differentials. Why can we do this?
The expression $\int 0.15B · dt$ cannot be evaluated directly because we do not know $B$ ’s relationship with $t$ . In other words, $B$ is defined implicitly in terms of $t$ . Therefore, alter the equation so that the $B$ and $dB$ are together and $t$ and $dt$ are together. Then integrate both sides.

To solve for $B$ , we need to “undo” the natural logarithm. How can we do this?
Why is an absolute value unnecessary?
Explain why $e^{0.15t + C}$ can be replaced with $Ce^{0.15t}$ ?
Examine the resulting equation for $B$ , the balance of Yong Li’s bank account after $t$ years. Where is the interest rate represented in this equation? What does the constant represent?
Explain why the general solution to Yong Li’s problem is $B = Ce^{0.15t}$ . Why does $B = e^{0.15t} + C$ not work? Use the derivative to justify your answer.
Write a differential equation for a bank account that grows with a simple annual interest rate of $35\%$ . Integrate implicitly. Then determine how many years it will take the initial balance to double.

$\boxed{\large\left. \begin{array}{l}{ \frac { d B } { d t } = 0.15 B }\\\\{ d B = 0.15 B \cdot d t }\\\\{ \frac { d B } { B } = 0.15 d t }\\\\{ \int \frac { d B } { B } = \int 0.15 d t }\\\\{ B = e ^ { 0.15 t } e ^ { C } }\\\\{ B = e ^ { 0.15 t } e ^ { C } }\\\\{ B = C e ^ { 0.15 t } e ^ { C } }\end{array} \right.}$

7-100.

Solve each of the following differential equations. Use implicit integration when necessary. Solve your equations for $y$ . Verify that your answers are correct by differentiation. Homework Help ✎

$\frac { d y } { d x }= 7x$

$\frac { d y } { d x }= 7y$

$\frac { d y } { d x }= e^{−y}$

7-101.

Nick and Roza are arguing over an implicit integration problem. Nick thinks the constant of integration must be added in when you find the antiderivative. Roza thinks you can add the $C$ later, after you have solved for $y$ . Which student is correct and why? Give an example to justify your answer. Homework Help ✎

7-102.

No calculator! Integrate. Homework Help ✎

$\int ( 3 e ^ { 4 x } + 2 ^ { x } ) d x$

$\int _ { 0 } ^ { \pi } \operatorname { sin } ^ { 3 } ( x ) \operatorname { cos } ( x ) d x$

$\int x ^ { 2 } ( 3 x ^ { 3 } + 5 ) ^ { 4 } d x$

$\int \frac { 5 x ^ { 0.25 } } { 4 x ^ { 1.25 } - 6 } d x$

$\int _ { e } ^ { e ^ { 3 } } \frac { 1 } { x } \operatorname { ln } ( x ) d x$

$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 - x ^ { 2 } } } d x$

7-103.

Population growth is proportional to the current population. For example, if a town has $100$ people, its change in population for a year will be based on those $100$ people. Therefore, its change will be much smaller than that of a large city like San Francisco, with population $800,000$ . Why? Write down your reasoning Homework Help ✎

7-104.

THE WEDDING CAKE, Part Three

Kiki is still trying to decide the shape of her wedding cake. She is now considering having $16$ layers! The largest diameter would be $16$ inches and each subsequent diameter would decrease by $1$ inch. Each layer of this cake would only be $1$ inch tall. Homework Help ✎

Setup a Riemann sum to calculate the volume of the cake.
Calculate the volume of this cake.

7-105.

Simplify these exponential expressions. Try each without a calculator first. Homework Help ✎

$e^{\ln(11)}$

$\ln(e^4)$

$\ln(e^e )$

$10^{\log(x)}$

7-106.

Multiple Choice: The average value of $f(x) = e^{\sin(x^2)}$ on the interval $−1 ≤ x ≤ 1$ is nearest to: Homework Help ✎

$−1.253$

$−1.414$

$1.253$

$1.414$

Calculus Third Edition

Solving Differential Equations