# Solving a Differential Equation: dy/dt = t/(y + 1)

• magnifik
In summary, the conversation is about finding the solution for the differential equation dy/dt = t/(y+1) with initial condition y(2) = 0. The solution process involves multiplying both sides by 2, completing the square, and applying the initial condition to find the value of C. There is also a discussion about isolating y and taking the square root.
magnifik
Can someone help me find the solution for this

dy/dt = t/(y + 1) with initial condition y(2) = 0

here's what i have done so far
(y + 1) dy = t dt
y2/2 + y = t2/2 + C

i am confused on how to isolate y.
this is the solution
without using the initial condition. where did the + 1 in the square root and - 1 outside of the square root come from?

First, multiply both sides by 2. Then complete the square on the LHS.

Don't forget, use y(2)=0, to evaluate C. (or c1).

After you've separated your variables and integrated both sides (which you have done correctly), multiply both sides by 2 to get rid of your coefficents and take the square root of both sides to solve for t. Then apply your condition.

TJ@UNF said:
After you've separated your variables and integrated both sides (which you have done correctly), multiply both sides by 2 to get rid of your coefficents and take the square root of both sides to solve for t. Then apply your condition.
Once you have multiplied both sides by 2, you will still have $y^2+ 2y$ on the left side and just "taking the square root" won't give you y. Complete the square as Char. Limit said in the very first response to this thread.

Thanks for the help!

## What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves one or more derivatives of an unknown function, and the goal is to find the function that satisfies the equation.

## What is the purpose of studying differential equations?

Differential equations are used to model real-world phenomena in many fields such as physics, engineering, economics, and biology. By understanding and solving these equations, we can make predictions, analyze data, and improve our understanding of various systems and processes.

## What are the types of differential equations?

There are several types of differential equations, including ordinary differential equations, partial differential equations, and stochastic differential equations. Ordinary differential equations involve a single independent variable, while partial differential equations involve multiple independent variables. Stochastic differential equations take into account randomness or uncertainty in the system.

## What is the difference between a linear and a nonlinear differential equation?

A linear differential equation is one in which the dependent variable and its derivatives appear only in a linear form. Nonlinear differential equations involve products, powers, or other nonlinear functions of the dependent variable and its derivatives. Linear differential equations are generally easier to solve than nonlinear ones, but they are limited in their ability to model complex systems.

## How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some common techniques include separation of variables, substitution, variation of parameters, and using special functions such as the Laplace transform. In some cases, numerical methods are used to approximate solutions. It is important to understand the properties of the equation and choose an appropriate method for solving it.

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