Calculus 1 : Equations
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Example Questions
Example Question #137 : Solutions To Differential Equations
Find the solution to the following differential equation:
Assume 
Using integrating factors:
Once we use this integrating factor, we can reduce the equation to:
Integrating both sides:
Example Question #391 : Equations
Determine which of the following are NOT solutions to the differential equation:
Since you may or may not know how to solve 
Let's start with 
If you try all the other function, they will solve the differential equation.
Example Question #137 : How To Find Solutions To Differential Equations
In circuits with a resistor, the equation for voltage drop is given by:
Given that 
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 
Example Question #391 : Equations
Find the derivative of the function 
The derivative of the trigonometric function 
Therefore, the correct answer is
Example Question #392 : Equations
Find the derivative of the function 
To find the derivative of the function, you must use the power rule
Using this rule, the correct answer
Example Question #142 : How To Find Solutions To Differential Equations
Find the derivative of the function 
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 #141 : Solutions To Differential Equations
Determine the solution to the following: 
The equation is not separable.
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 #144 : How To Find Solutions To Differential Equations
Solve the following differentiable equation:
with a condition of 
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
If an equation is homogenous you can do a substitution of 
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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