Separable method is a powerful technique used to solve a wide range of differential equations. In this note, we will discuss what a separable equation is, how to solve it, and what are some of its applications.
A separable differential equation is a differential equation that can be written in the form:
dy/dx = f(x)g(y
where f(x) and g(y) are functions of x and y, respectively.
The idea behind the separable method is to rewrite the differential equation so that all the y terms are on one side and all the x terms are on the other side. Then, we can integrate both sides to obtain the solution.
To solve a separable equation, we follow the following steps:
The constants of integration can be determined by using initial or boundary conditions or by matching the solution to a particular form.
Separable equations are used to model a variety of phenomena in physics, chemistry, biology, and economics. Some examples include:
Overall, the separable method is a powerful tool for solving differential equations and has many applications in science and engineering.
22. Solve (e^x+e^{-x})dy/dx=y^2 be separable method To solve the differential equation by separable method, we need to rearrange it so that all the terms are on one side and all the terms are on the other side. Then, we can integrate both sides to obtain the solution. First, we can multiply both sides by and … Read more
21. Solve be separable method To solve the differential equation by separable method, we need to rearrange it so that all the terms are on one side and all the terms are on the other side. Then, we can integrate both sides to obtain the solution. First, we can divide both sides by to get … Read more
20. Solve dy/dx={xy+2y-x-2}/{xy-3y+x-3} by separable method. To solve the differential equation we can rearrange the terms in the numerator and denominator to get: Now we can use the method of separation of variables by multiplying both sides by the denominator and then separating the variables as follows: … Read more
19. Solve dy/dx={xy+3x-y-3}/{xy-2x+4y-8} by separable method. To solve the differential equation we can rearrange the terms in the numerator and denominator to get: Now we can use the method of separation of variables by multiplying both sides by the denominator and then separating the variables as follows: … Read more
18. dN/dt+N=Nte^{t+2} by separable method. To solve the differential equation by separable method, we need to separate the variables and on opposite sides of the equation and integrate both sides. Starting with the given differential equation: We can first solve the homogeneous equation to get the complementary solution: … Read more
17. dP/dt=P-P^2 by separable method. To solve the differential equation by separable method, we need to separate the variables and on opposite sides of the equation and integrate both sides. Starting with the given differential equation: We can divide both sides by to obtain: Now we can integrate both sides with … Read more
16. dQ/dt=k(Q-70) solve by separable method. To solve the differential equation by separable method, we need to separate the variables and on opposite sides of the equation and integrate both sides. Starting with the given differential equation: We can divide both sides by to obtain: Now we can integrate both sides … Read more
\frac{dS}{dr}=kS solve by separable method. To solve the differential equation by separable method, we need to separate the variables and on opposite sides of the equation and integrate both sides. Starting with the given differential equation: We can divide both sides by to obtain: Now we can integrate both sides with … Read more
14. x(1+y^2)^{\frac{1}{2}}dx=y(1+x^2)^{\frac{1}{2}}dy solve separable method. To solve this differential equation by separable method, we need to separate the variables x and y on opposite sides of the equation and integrate both sides. Starting with the given differential equation: (1) We can divide both sides by to obtain: (2) Now we can integrate both … Read more
Solve Separable Differential Equation (e^y + 1)^2e^(-y)dx + (e^x + 1)^2e^(-x)dy = 0 To solve this differential equation using the separable method, we need to write it in the form of where and are functions of only and , respectively. Starting with the given equation: We can divide both sides by to get: … Read more
