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 #61 : Differential Equations

Find the solution of the separable equation:

Possible Answers:

Correct answer:

Explanation:

We begin by factoring out the y term on the right side, then dividing it over and multiplying the x differential over:

We then integrate both sides in the proper direction to get:

To be able to flip the right side of the equation it is necessary to find a common denominator for each term, so we get:

We then multiply by negative 1 and flip the equation.

Example Question #62 : Differential Equations

Find the solution to the homogeneous first order differential equation:

Possible Answers:

Correct answer:

Explanation:

Homogeneous first order differential equations can always be solved by brute force using the method of an integrating factor. However, this differential equation is simple enough to use a different method. First, we set the derivative equal to the second term:

Then we think: The only way that a derivative can be equal to a scalar multiplication of the original function is if the derivative does not change the change the function in any way other than multiplying it by some constant. The only function that holds this property is the exponential function. Then, we can assume that

 for some r.

Then, we differentiate this function and substitute the values into the differential equation

.

Example Question #301 : Equations

Find the solution to the separable equation.

Possible Answers:

Correct answer:

Explanation:

We begin by separating the equation:

We then integrate each side of the equation using the power rule,

 and special case that  to get,

.

We then exponentiate to get

.

Example Question #61 : Differential Equations

Find the derivative of .

Possible Answers:

Correct answer:

Explanation:

Looking over our problem, we have a composite function , where  and . We are trying to find the derivative of , or , which by the Chain Rule can be defined as 

Therefore:

Example Question #65 : Differential Equations

Find the general solution to the following differential equation:

Possible Answers:

Correct answer:

Explanation:

The differential equation  can be rewritten as .

The next step is to multiply by  and divide by  to get the x terms and the y terms on the same side as  and , respectively.

Next, integrate both sides of the equation  

 .

The rules used for the integration are

, where n=2 in this example, and 

The solution of the integration is as follows: 

.

After, both sides of the equation are exponentiated using e (to remove the natural log): 

.

To simplify,  can be rewritten as .  

Then, to isolate y, subtract 3 and divide both sides by 2:

 .

Note that  and were  changed to just C, because they are themselves another constant of integration. 

Example Question #302 : Equations

Find the particular solution at  to the given differential equation:

Possible Answers:

Correct answer:

Explanation:

 To solve the differential equation, you first seperate the x and y variables: 

.

Then, integrate on both sides: 

 which is equal to: .

This integration was performed using the rules

 and .

(The constant 2 had no part in the integration and was therefore unchanged.) Note that there is only one constant of integration, C, in the entire equation. This is the general solution of the differential equation. We want the particular solution at the point , though. So, you plug in that point into the equation to solve for C: 

. The natural log of one is zero, so .

The final answer, which doesn't contain C, is .

Example Question #51 : How To Find Solutions To Differential Equations

What is the second derivative of ?

Possible Answers:

Correct answer:

Explanation:

The original function can be written as

 

Taking the first derivative using chain rule and simplifying gives us

 

Now, to take the second derivative, we may either use the product rule or the quotient rule. The product rule is used arbitrarily. Either of these methods will also implement the chain rule on the  term.

 

Simplying and finding a common denominator gives us

Example Question #2391 : Calculus

What is the derivative of ?

Possible Answers:

Correct answer:

Explanation:

In the problem above, we can consider  to be a composite function , where  and .  

According to the Chain Rule, .

Applying this rule to our composite function:

Example Question #51 : Solutions To Differential Equations

What is the derivative of ?

Possible Answers:

None of the above

Correct answer:

Explanation:

In the problem above, we can consider  to be a composite function , where  and . .  

According to the Chain Rule, .

Applying this rule to our composite function:

Example Question #64 : Differential Equations

What is the derivative of ?

Possible Answers:

None of the above

Correct answer:

Explanation:

In the problem above, we can consider  to be a composite function , where  and .  

According to the Chain Rule, .

Applying this rule to our composite function:

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