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.

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

.

We begin by separating the equation:

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

and special case thatto get,

.

We then exponentiate to get

.

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

Therefore:

The differential equationcan be rewritten as.

The next step is to multiply byand divide byto get the x terms and the y terms on the same side asand, 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:

.

之后,方程两边取幂d 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 thatandwere changed to just C, because they are themselves another constant of integration.

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.

The original function can be written as

Taking the first derivative using chain rule and simplifying gives us

现在,二阶导数,我们可以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 theterm.

Simplying and finding a common denominator gives us

In the problem above, we can considerto be a composite function, whereand.

According to the Chain Rule,.

Applying this rule to our composite function:

.

In the problem above, we can considerto be a composite function, whereand. .

According to the Chain Rule,.

Applying this rule to our composite function:

.

In the problem above, we can considerto be a composite function, whereand.

According to the Chain Rule,.

Applying this rule to our composite function:

.