# Calculus 1 : Solutions to Differential Equations

## Example Questions

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### Example Question #1 : How To Find Solutions To Differential Equations

Find the derivative of (5+3x)5.

5(5+3x)^4
15(5+3x)^4
5(5+3x)^4x
5x(5+3x)^4
15x(5+3x)^4
Explanation:

We'll solve this using the chain rule.

Dx[(5+3x)5]

=5(5+3x)4 * Dx[5+3x]

=5(5+3x)4(3)

=15(5+3x)4

### Example Question #1 : How To Find Solutions To Differential Equations

Find Dx[sin(7x)].

7cos(7x)
7sin(7x)cos(7x)
-7sin(7x)
-7cos(7x)
7sin(7x)
Explanation:

First, remember that Dx[sin(x)]=cos(x). Now we can solve the problem using the Chain Rule.

Dx[sin(7x)]

=cos(7x)*Dx[7x]

=cos(7x)*(7)

=7cos(7x)

### Example Question #1 : Solutions To Differential Equations

Calculate fxxyz if f(x,y,z)=sin(4x+yz).

-16cos(4x+yz) +16yzsin(4x+yz)
-16sin(4x+yz)
arctan(4x+yz)
cos(4x+yz)
4sin(4x+yz)
Explanation:

We can calculate this answer in steps.  We start with differentiating in terms of the left most variable in "xxyz".  So here we start by taking the derivative with respect to x.

First, fx= 4cos(4x+yz)

Then, fxx= -16sin(4x+yz)

fxxy= -16zcos(4x+yz)

Finally, fxxyz= -16cos(4x+yz) + 16yzsin(4x+yz)

Integrate

Explanation:

thus:

Integrate :

Explanation:

thus:

### Example Question #1 : How To Find Solutions To Differential Equations

Find the general solution, , to the differential equation

.

Explanation:

We can use separation of variables to solve this problem since all of the "y-terms" are on one side and all of the "x-terms" are on the other side.  The equation can be written as .

Integrating both sides gives us

### Example Question #17 : Differential Equations

Consider  ; by multiplying by  both the left and the right hand sides can be swiftly integrated as

where .  So, for example,  can be rewritten as:

. We will use this trick on another simple case with an exact integral.

Use the technique above to find  such that  with  and .

Hint: Once you use the above to simplify the expression to the form , you can solve it by moving  into the denominator:

Explanation:

As described in the problem,  we are given

.

We can multiply both sides by :

Recognize the pattern of the chain rule in two different ways:

This yields:

We use the initial conditions to solve for C, noticing that at  and  This means that C must be 1 above, which makes the right hand side a perfect square:

To see whether the + or - symbol is to be used, we see that the derivative starts out positive, so the positive square root is to be used. Then following the hint we can rewrite it as:

,

which we learned to solve by the trigonometric substitution, yielding:

Clearly  and the fact that  again gives us  so

### Example Question #2 : How To Find Solutions To Differential Equations

What are all the functions  such that

?

for arbitrary constants k and C

for arbitrary constants k and C

for arbitrary constants k and C

for arbitrary constants k and C

for arbitrary constants k and C

for arbitrary constants k and C

Explanation:

Integrating once, we get:

Integrating a second time gives:

We integrate the first term by parts using  to get:

Canceling the x's we get:

Defining  gives the above form.

### Example Question #2341 : Calculus

The Fibonacci numbers are defined as

and are intimately tied to the golden ratios , which solve the very similar equation

.

The n'th derivatives of a function are defined as:

Find the Fibonacci function defined by:

whose derivatives at 0 are therefore the Fibonacci numbers.

Explanation:

To solve , we ignore  of the derivatives to get simply:

This can be solved by assuming an exponential function , which turns this expression into

,

which is solved by  . Our general solution must take the form:

Plugging in our initial conditions  and , we get:

### Example Question #20 : Differential Equations

Find the particular solution given

Explanation:

The first thing we must do is rewrite the equation:

We can then find the integrals:

The integrals as as follows:

we're left with

We then plug in the initial condition and solve for

The particular solution is then:

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