1st order differential equation

In summary, the general solution of the first-order differential equation is y=\pm\sqrt{x^{2}-2+e^{C}} with the boundary condition e^{C}=2.
  • #1
subzero0137
91
4
Find the general solution of the first order differential equation [itex](y+x^{2}y)\frac{dy}{dx}=3x+xy^{2}[/itex], with [itex]y(1)=1[/itex].



My attempt:
[tex]\frac{y}{3+y^{2}}dy=\frac{x}{1+x^{2}}dx ∴ \frac{1}{2}\int \frac{2y}{3+y^{2}}dy=\frac{1}{2}\int \frac{2x}{1+x^2}dx[/tex]
[tex]=\frac{1}{2}ln|3+y^{2}|=\frac{1}{2}ln|1+x^{2}|+C[/tex] ∴ [tex]y^{2}+3=x^{2}+1+e^{C}[/tex] ∴ [tex]y=\pm\sqrt{x^{2}-2+e^{C}}[/tex] Applying boundary conditions gives [tex]1=\pm\sqrt{1^{2}-2+e^{C}} \Rightarrow e^{C}=2[/tex] Therefore [tex]y=\pm x[/tex].


Is this right?
 
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  • #2
subzero0137 said:
[tex]y^{2}+3=x^{2}+1+e^{C}[/tex]

Not quite right here. Pay particular attention to your exponential rules. Remember that

[tex]a^{b+c}=a^ba^c[/tex]
 
  • #3
Mentallic said:
Not quite right here. Pay particular attention to your exponential rules. Remember that

[tex]a^{b+c}=a^ba^c[/tex]

Ohhh! Of course...silly me.
 
  • #4
subzero0137 said:
Ohhh! Of course...silly me.

Don't worry, I answered a whole homework problem set on this topic with the exact same mistake in each and every question haha :biggrin:
 

What is a 1st order differential equation?

A 1st order differential equation is an equation that involves a function and its first derivative. It expresses the rate of change of a variable in terms of the variable itself.

What is the general form of a 1st order differential equation?

The general form of a 1st order differential equation is dy/dx = f(x,y), where y is the dependent variable and x is the independent variable. The function f(x,y) relates the values of y and its derivative dy/dx at any given point.

How do you solve a 1st order differential equation?

There are several methods to solve a 1st order differential equation, including separation of variables, integrating factors, and using exact equations. Each method involves manipulating the equation to isolate the dependent and independent variables, and then integrating to find the general solution.

What is the difference between an ordinary and a partial 1st order differential equation?

An ordinary 1st order differential equation involves only one independent variable, while a partial 1st order differential equation involves multiple independent variables. The solution to a partial differential equation is a function of more than one variable, while the solution to an ordinary differential equation is a function of only one variable.

What are some real-world applications of 1st order differential equations?

1st order differential equations have many applications in physics, engineering, economics, and biology. They are used to model the rate of change of physical systems, such as population growth, chemical reactions, and electric circuits. They are also used to solve problems involving motion, heat transfer, and fluid mechanics.

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