How to Solve This Differential Equation Analytically?

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  • #1
SantiagoCR
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calculate integral of a differential equation
Hello,

can someone help me to solve the following differential equation analitically:

$$\frac{2 y''}{y'} - \frac{y'}{y} = \frac{x'}{x}$$

where ##y = y(t)## and ##x = x(t)##

br

Santiago
 
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  • #2
Hint: $$\frac{2y''}{y'}=2\log\left(y'\right)',\quad\frac{y'}{y}=\log\left(y\right)',\quad\frac{x'}{x}=\log\left(x\right)'$$
 
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Likes Kumail Haider, SantiagoCR, Frabjous and 1 other person
  • #3
renormalize said:
Hint: $$\frac{2y''}{y'}=2\log\left(y'\right)',\quad\frac{y'}{y}=\log\left(y\right)',\quad\frac{x'}{x}=\log\left(x\right)'$$
cool, thank you very much!
 

FAQ: How to Solve This Differential Equation Analytically?

What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It describes how a quantity changes in relation to another quantity, often representing physical phenomena such as motion, heat, or population dynamics.

What are the different types of differential equations?

Differential equations can be categorized into several types, including ordinary differential equations (ODEs), which involve functions of a single variable, and partial differential equations (PDEs), which involve functions of multiple variables. They can also be classified as linear or nonlinear, homogeneous or non-homogeneous, and of various orders based on the highest derivative present.

What methods can be used to solve differential equations analytically?

Common methods for solving differential equations analytically include separation of variables, integrating factors, characteristic equations for linear ODEs, and the method of undetermined coefficients. For PDEs, techniques like separation of variables, Fourier series, and transform methods may be employed.

What is the importance of initial or boundary conditions in solving differential equations?

Initial or boundary conditions are crucial for finding a unique solution to a differential equation. They specify the values of the function or its derivatives at specific points, allowing us to determine a particular solution from the general solution that may contain arbitrary constants.

How can I verify that my solution to a differential equation is correct?

You can verify your solution by substituting it back into the original differential equation to see if it satisfies the equation. Additionally, checking that the solution meets any initial or boundary conditions provided can further confirm its correctness.

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