Need help solving a differential equation

In summary, the conversation discusses a differential equation problem involving separation of variables. The attempt at a solution involves finding the function y and y' and then using the initial condition to solve for the constants of integration. However, a mistake is made when dividing by x, as the initial condition is for x=0. This needs to be taken into consideration when solving the problem.
  • #1
Planefreak
10
0

Homework Statement



xy' - (x+1)y = 0, y(0) = 5

Homework Equations



Well all of the differential equation stuff seems relevant to me. But from the beginning dy/dx + P(x)y = Q(x) and yeintegral(P(x)dx) = integral(Q(x)eintegral(P(x)dx)dx + C. The separation of variables M(x)dx + N(y)dy = 0.

The Attempt at a Solution



Well I tried following separating everything out: xy' - (x+1)y = 0 y' - ((x+1)/x)y = 0/x then p(x) = -(x+1)/x and eintegral(P(x)dx) = e-integral((x+1)/x dx) and using some more math we end up with e-x/x and that is where I fall apart. Do I integrate this new function and then what? I come up with x/e-x for y and (x-1)ex for y' but when I put it back into the original equation i get x2 + x = x2 -x. Where did I go wrong?
 
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  • #2
Be careful...you're dividing by x but the initial condition is for x=0. You need to consider the function on an open interval about x=0 and so you cannot divide by x.
 

What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves the use of derivatives to model the rate of change of a system over time.

Why do we need to solve differential equations?

Differential equations are used to model real-world systems, such as population growth, chemical reactions, and the motion of objects. Solving these equations helps us understand how these systems change over time and make predictions about their behavior.

What are the methods for solving differential equations?

There are various methods for solving differential equations, including separation of variables, substitution, and using integrating factors. The choice of method depends on the type and complexity of the equation.

What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve a single independent variable, while partial differential equations involve multiple independent variables. Ordinary differential equations also have a single solution, while partial differential equations may have multiple solutions.

What are some common applications of differential equations?

Differential equations are used in many fields, including physics, engineering, biology, economics, and finance. They are particularly useful in modeling dynamic systems and predicting their future behavior.

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