Calculus 1 : How to find solutions to differential equations

Study concepts, example questions & explanations for Calculus 1

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Example Questions

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Example Question #395 : Equations

Find the derivative of the function .

 

Possible Answers:

Correct answer:

Explanation:

To find the derivative of the function, you must use the power rule 

.

Using this rule, the correct answer 

 is obtained.

Example Question #1441 : Functions

Find the derivative of the function .

Possible Answers:

Correct answer:

Explanation:

To take the derivative of the first two terms, you use the power rule 

.

To take the derivative of sin(x), there is the identity that,

 .

Using all of the rules and applying, 

.

Example Question #397 : Equations

Determine the solution to the following:  with .

Possible Answers:

The equation is not separable.  

Correct answer:

Explanation:

Solving a differential equation tends to be quite simple if it is separable. A separable equation is one that can be rewritten in the following form 

.  

From here we can manipulate the equation 

and then you can integrate both sides.  

We can rewrite the equation as 

 

and we therefore see that it is separable.  

Next we manipulate it to get the the y terms on the left hand side and the x terms on the right hand side like the following 

.  

The next step is to integrate both sides 

.  

This gives us 

,

since the c's are just arbitrary constants we can combine them to get 

.  

Since we want to write this as an equation in terms of y we will apply e to both sides.  

Therefore we get 

.  

Since the problem gives us an initial condition we solve for c.  

Therefore 

 

so c=4.  

Thus the solution is 

.

Example Question #398 : Equations

Solve the following differentiable equation: 

 

with a condition of .

Possible Answers:

Correct answer:

Explanation:

When given a first order differential equation to solve, one of the first things to check is that it is homogenous.Homogenous equations are ones in which, given

 given any t.

If an equation is homogenous you can do a substitution of  and this new equation will be separable.  

The first step is to check if the following equation is homogenous, 

.  

Next we show that the equation is homogenous

.

 So we now  replace y with xz. This gives us the following new equation:

.  

This is a separable equation which we manipulate and integrate both sides 

 

which leads to 

.  

Thus we get the equation 

,

then plugging y back in we get 

.  

So our family of equations is 

 

yet plugging in the given condition gives us that c=0.  

Thus the final solution 

.

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