Equations with separable variables

where p ( x ) and h ( y ) are continuous functions.

Considering the derivative y’ as a ratio of differentials , we move dx to the right-hand side and divide the equation by h ( y ) :

Of course, we need to make sure that h ( y ) ≠ 0 . If there is a number x ₀ such that h ( x ₀ ) = 0, then this number will also be a solution to the differential equation. Dividing by h ( y ) results in the loss of the indicated solution.

Denoting , we write the equation in the form:

Now the variables are separated, and we can integrate the differential equation:

where C is the constant of integration.

Calculating the integrals, we obtain the expression.

describing the general solution of an equation with separable variables.

In this case, p ( x ) = 1 and h ( y ) = y ( y +2) . We divide the equation by h ( y ) and move dx to the right side:

      

Note that when dividing, we could lose the solutions y = 0 and y = −2 in the case where h ( y ) is equal to zero. Indeed, let us verify that y = 0 is a solution to this differential equation. Let

      

Substituting this into the equation yields: 0 = 0. Therefore, y = 0 will be one of the solutions. Similarly, we can verify that y = -2 is also a solution to the equation.

Let’s return to the differential equation and integrate it:

      

The integral on the left-hand side can be calculated using the method of undetermined coefficients:

      

Thus, we obtain the following expansion of the rational fraction in the integrand:

      

Hence,

      

Let’s rename the constant: 2 C = C ₁ . As a result, the final solution to the equation is written as:

      

The general solution here is expressed implicitly. In this example, we can transform it and obtain an explicit answer as a function y = f ( x, C ₁ ), where C ₁ is a constant. However, this cannot be done for all differential equations.

Let us write this equation in the following form:

      

Divide both sides by ( x ² + 4) y :

      

Clearly, x ² + 4 ≠ 0 for all real x. Let’s check that y = 0 is one of the solutions to the equation. After substituting y = 0 and dy = 0 into the original differential equation, it is clear that the function y = 0 is indeed a solution to the equation.

Now we can integrate the resulting equation:

      

Note that dx ² = d ( x ² + 4) . Therefore,

      

Let us represent the constant C as ln C1, where C ₁ > 0. Then

      

Thus, the given differential equation has the following solutions:

      

The answer we obtained can be simplified. In fact, we introduce an arbitrary constant C, taking values ​​from −∞ to +∞. Then the solution can be written as:

      

When C = 0, it becomes equal to y = 0.

We transform the equation as follows:

      

Dividing by e ^y does not lead to a loss of solutions, since e ^y > 0. After integration, we obtain

      

This answer can be expressed explicitly:

      

In the last expression, the constant C is assumed to be > 0 in order to satisfy the domain of the logarithmic function.

Let’s divide both sides of the equation by x :

      

We assume that x ≠ 0, since the domain of the original equation is the set x > 0.

As a result of integration, we obtain:

      

The integral on the right-hand side is calculated as follows:

      

Therefore, the general solution in implicit form is:

      

where C ₁ = 2 C is the constant of integration.

Let us now find the value of C ₁ that satisfies the initial condition y (1) = −1 :

      

Thus, a particular solution of a differential equation with a given initial condition (the Cauchy problem) is described by an algebraic equation:

      

Let us write this equation in the following form:

      

Divide both sides by tan y cot ² x :

      

Let’s check if we’ve lost any solutions as a result of the division. We need to examine the following two roots:

      

Substituting into the original equation, we see that is the solution to the equation.
The second possible solution is described by the formula.

      

Here we get the answer:

      

which does not satisfy the original differential equation.

Now we can integrate the differential equation and find its general solution:

      

The final answer is written as:

      

Let’s rewrite the equation as follows:

      

Divide both sides by 1 + e ^x :

      

Since 1 + e ^x > 0, we didn’t lose any solutions during the division. We integrate the resulting equation:

      

Now we find the constant C from the initial condition y (0) = 0.

      

Therefore, the final answer is:

      

The product of xy in each part does not allow for separation of variables. Therefore, we will make a substitution:

      

The relationship for differentials is:

      

Substituting this into the equation, we get:

      

Next, multiplying both parts of x, we can write, after the appropriate abbreviations:

      

Let’s take into account that x = 0 is a solution to the equation (this can be verified by direct substitution).

The last expression can be simplified somewhat:

      

Now the variables x and t are separated:

      

As a result of integration, we find:

      

By performing the back substitution t = xy, we obtain the general solution of the differential equation:

      

The full answer is written as:

      

Let’s use the following substitution:

      

As a result, the equation takes the form:

      

Hence,

      

Let’s integrate the last equation:

      

Since u = x + y, the final answer is implicitly written as: