Calculus 1 : How to find solutions to differential equations

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #381 : Equations

Find the general solution to the following differential equation:

Possible Answers:

Correct answer:

Explanation:

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Example Question #382 : Equations

 

 

Find .

Possible Answers:

 

Correct answer:

 

Explanation:

To find  first find .

Since,

then,

.

We have to use the chain rule and power rule to solve this problem:

Chain Rule: 

Power Rule: 

Therefore,

Example Question #383 : Equations

Solve for  at .

Possible Answers:

Correct answer:

Explanation:

To solve this problem, we must first find f'(t) by deriving the original equation.

In this case, we just need to use the power rule which states,

We can then derive f'(t), again using just the power rule which states .

When we plug 4 into this final function, we arrive at the answer:

Example Question #1431 : Functions

Finding where the first derivative is positive tells us what about an equation?

Possible Answers:

Where the function is decreasing.

Where the function is concave up.

Where the function is concave down.

Where the function is increasing.

Correct answer:

Where the function is increasing.

Explanation:

Taking the first derivative gives us the function for the slope for our entire equation. Thus, finding where the slope is positive tells us where the function is increasing.

Example Question #132 : How To Find Solutions To Differential Equations

Find the general solution to the following differential equation:

Possible Answers:

Correct answer:

Explanation:

To find the general solution for the separable differential equation, we must move x and dx, y and dy to the same sides:

Next, we integrate both sides:

The integrals were found using the following rules:

To solve for y, we must exponentiate both sides of the equation:

Example Question #383 : Equations

Find the general solution for the following differential equation:

Possible Answers:

Correct answer:

Explanation:

To solve the separable differential equation, we must move x and dx, y and dy to their respective sides, and integrate both sides:

The integrations were performed using the following rules:

The equation then becomes

Note that we combine all constants of integration to a single C.

Now, we solve for y by taking the natural logarithm of both sides of the equation:

Simplifying, we get

.

 

 

 

Example Question #391 : Equations

Find the solution to the following differential equation:

Assume  is a function of 

Possible Answers:

Correct answer:

Explanation:

Using integrating factors:

 

Once we use this integrating factor, we can reduce the equation to:

Integrating both sides: 

Example Question #392 : Equations

Determine which of the following are NOT solutions to the differential equation:

Possible Answers:

Correct answer:

Explanation:

Since you may or may not know how to solve  order Differential equations, you can always plug in the answers and see if it is correct. 

Let's start with 

If you try all the other function, they will solve the differential equation. 

Example Question #393 : Equations

In circuits with a resistor, the equation for voltage drop is given by:

, where  is voltage,  is charge, and  is resistance. 

Given that , and , determine the voltage difference from  at  

Possible Answers:

Correct answer:

Explanation:

This involves solving a differential equation for 

Firstly, we determine the value of :

At 

This means our equation is now:

To determine the difference of  from , we take the definite integral:

Example Question #394 : Equations

Find the derivative of the function  

Possible Answers:

Correct answer:

Explanation:

The derivative of the trigonometric function  is .

Therefore, the correct answer is 

.

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