Need help finding solutions to Diff eq

• primus
In summary, to solve the first equation, you need to find a function u(x) so that 3u(x)y'=3u(x)y-2u(x)y, and to solve the second equation, you need to find a function u(x) so that 2u(x)y'+3u(x)y=4.

primus

Hi, I need help to solve two differential equation:

1. Find all solutions to the differential equation 3y' - 2y = 1 - x

2. Find the solution to 2y' + 3y = 4 when y(2) = 0

I would be happy if anyone could explain the general rule to solve these two equations, because the book I use only show poor examples.

primus said:
Hi, I need help to solve two differential equation:

1. Find all solutions to the differential equation 3y' - 2y = 1 - x

2. Find the solution to 2y' + 3y = 4 when y(2) = 0

I would be happy if anyone could explain the general rule to solve these two equations, because the book I use only show poor examples.

Hello Primus,

These are first-order ODEs: The technique is to find an integrating factor, multiply both sides by it, then integrate. The first case, integrate over a general interval ($x_o$,x), in the second case, integrate over a definite interval (2,x). However, you may not be familiar with these operations. I'll give you some time to look into them on your own. If you still have questions, ask, and me or someone else will work through them with you.

Integrating factor:
In problem 1, find a function u(x) so that 3(u(x)y(x))'= 3u(x)y'- 2u(x)y
Multiply the entire equation by u(x) and the left side is easy to integrate.

In problem 2, find a function u(x) so that 2(u(x)y(x))'= 2u(x)y'+ 3u(x)y.

Linear nonhomogenous constant coeff.I-st order ODE-s.

1.Solve the homogenous equation bby separating variables.
2.Find the particular solution to the nonhomogenous equation (through Lagrange's method).
3.Write the general solution & impose the initial condition (for the second which is a Cauchy problem).

Daniel.

Well, here's the first step:

Whenever you have a first order ODE in that form, place it in standard form:

y'+A(x)y=B(x)

Thus, you have:

$$y^{'}-\frac{2}{3}y=\frac{1}{3}(1-x)$$

Once you have it in standard form, the integration factor is e raised to the integral of A(x):

$$\mu(x)=e^{\int -\frac{2}{3}dx}=e^{-\frac{2}{3}x$$

Now, just multiply both sides of the standard form equation by the intgration factor:

$$e^{-\frac{2}{3}x}(y'-\frac{2}{3}y)=\frac{1}{3}e^{-\frac{2}{3}x}(1-x)$$

The left side is the differential of $e^{-2/3 x}y$ so when you integrate that, you're left with just $e^{-2/3 x}y$ right?

Just integrate indefinitely both sides now and remember to add a constant of integation. It's easier than figuring it by integrating from $x_0$ to x.

1. What is a differential equation?

A differential equation is an equation that involves a function and its derivatives. It describes the relationship between a function and its rate of change.

2. Why are differential equations important in science?

Differential equations are used to model and solve real-world problems in various scientific fields such as physics, engineering, economics, and biology. They help us understand and predict how systems change over time.

3. What are the different types of differential equations?

The three main types of differential equations are ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve a single independent variable, while PDEs involve multiple independent variables. SDEs incorporate randomness into the equation.

4. How can I find solutions to differential equations?

There are several methods for solving differential equations, such as separation of variables, substitution, and using integrating factors. It is also common to use software programs or numerical methods to approximate solutions.

5. What are some common applications of differential equations?

Differential equations are used in a wide range of applications, including modeling population growth, predicting the motion of objects in physics, analyzing circuits in electrical engineering, and studying chemical reactions in chemistry. They also have applications in financial modeling and image processing.