# Calculus 1 : How to find solutions to differential equations

## Example Questions

### Example Question #81 : Solutions To Differential Equations

Find the general solution to the differential equation:

Explanation:

To solve the seperable differential equation, we must put the x and dx and y and dy on the same sides:

Now, integrate both sides:

The integrations were performed using the following rules:

Finally, solve for y:

Note that the Cs combined to make one constant of integration.

### Example Question #91 : Differential Equations

Find the general solution to the differential equation:

Explanation:

The solution for the separable differential equation can be found by first separating x and dx, y and dy:

Now, integrate both sides:

The following rules were used for integration:

Finally, solve for y:

Note that the Cs combined to make one constant of integration.

### Example Question #92 : Differential Equations

Find the derivative of the function

.

Explanation:

To find this derivative, use of both the product rule and quotient rule for derivatives will be necessary. The latter states:

For our function

The  derivative is simply

For the  derivative, the product rule will be useful:

Where

And the derivative is:

Putting all of this together, the derivative of f(x) is:

### Example Question #93 : Differential Equations

Find the general solution to the given differential equation:

Explanation:

To solve the differential equation, we must move the x and y terms with dx and dy, respectively:

Now we can integrate:

using the following rules:

To finish, write the equation in terms of y alone:

### Example Question #81 : Solutions To Differential Equations

Find the derivative of

.

Explanation:

This can be easily separated into two derivatives added together:

The second function is easy: the derivative of any constant is 0. But for the first, we must use the chain rule.

Recall:

Our outside function is  and our inside function is .

So the chain rule tells us we must take the derivative of  and plug  into that function.

The derivative of  is , so we have

.

Now all we have to do is find the derivative of , which we know is .

.

### Example Question #85 : How To Find Solutions To Differential Equations

Find the solution to the differential equation.

Explanation:

By the power rule, we know that

, where  are constants and  is a variable.

In our case,

, where  is a constant.

### Example Question #86 : How To Find Solutions To Differential Equations

Which type of function solves the differential equation given.

Rational

Logarithmic

Exponential

Polynomial

Polynomial

Explanation:

Without doing very much, we can see that the solution to this differential equation will be such that the derivative is equal to the original function divided by

One such type of function is a polynomial, where the derivative decreases the order of the function by , which is equivalent to dividing by .

We can also show this through integration.

First separate the variables.

Now take the integral remembering the rule for natural logs.

Thus we get,

From here recall the properties of natural logs such that .

Therefore we can rewrite our function and exponentiate it to solve for y.

.

From here, we can see that it is a polynomial.

### Example Question #87 : How To Find Solutions To Differential Equations

Find the general solution to the differential equation:

Explanation:

To find the general solution to the seperable differential equation, we must seperate the x and dx, y and dy to seperate sides:

Now, integrate both sides (note that the two constants of integration combine to make one C):

The following rules were used for integration:

Finally, solve for y:

### Example Question #89 : Solutions To Differential Equations

Where will  have a local minimum or maximum between  and

,

,

Explanation:

We must find the maximums and minimumss by setting the derivative equal to zero.

The derivative of  is found using the chain rule,

and the trigonometric rule of derivatives for cosine,

.

Applying the above rules we find the derivative to be,

when , so in this case, we must divide each angle by two since we are dealing with , so .

### Example Question #81 : How To Find Solutions To Differential Equations

Which one of the following functions is continuous and has continuous derivative at 2?

Explanation:

In order for the graph to be continuous, both of the parts of the piecewise function must be equal.

For the derivative to be equal, one must take the derivatives at each part and see if this new piecwise function is equal at 2 as well.

Only

fufills both these requirements.

Remember to use the power rule to find the derivatives of the piecewise function, .