Struggling with ODE: Find Particular Solution y(0)=1

[wozzers]

i am having issues solving an ODE it is given as

y'= (1-2y-4x)/(1+y+2x)

I've been told to find the particular solution when y(0)=1

please help

 

[Mandelbroth]

I am having issues solving an ODE it is given as

y'= (1-2y-4x)/(1+y+2x)

I've been told to find the particular solution when y(0)=1

please help

Your reason for editing was adorable. :tongue:

So, let's look at this.

We have that $\frac{dy}{dx} = \frac{1-2y-4x}{1+y+2x}$. We need to reduce this to something we know how to solve. Do you see any useful substitutions?

 

[wozzers]

Your reason for editing was adorable. :tongue:

So, let's look at this.

We have that $\frac{dy}{dx} = \frac{1-2y-4x}{1+y+2x}$. We need to reduce this to something we know how to solve. Do you see any useful substitutions?

at the moment none i thought about doing something with the y+2x and using a u substitution but noticed the signs were different so i am just lost currently and i wasn't sure if that is even applicable in this situation

 

[lurflurf]

^That is good

u=y+2x+1 is even better

 

[wozzers]

^That is good

u=y+2x+1 is even better

okay i took the derivative of y+2x+1 and got 3dydx is that right or a little off with the dydx from there i am lost once again

 

[lurflurf]

^
d(y+2x+1)=dy+2dx
or
(y+2x+1)'=y'+2

try to write the equation in terms of y+2x+1

 

[Ackbach]

In general, if you have $y'=L_{1}(x,y)/L_{2}(x,y),$ where $L_{1},L_{2}$ are linear functions of $x$ and $y$, then your substitution depends on how the two lines relate to each other. Tenenbaum and Pollard have a great section in their book Ordinary Differential Equations, where they show you exactly which substitution to use for the following three cases:

Case 1: They are the same line.
Case 2: The two lines intersect at one point.
Case 3: The two lines are parallel.

 

[wozzers]

i will take a look at the book ackbeet as i know this is something i am going to have to apply consistently to completely comprehend this type of problem

 

[HallsofIvy]

There is no real need to look for substitutions- it's pretty simple as is. This is an "exact" equation.

$$\frac{dy}{dx}= \frac{1- 2y- 4x}{1+ y+ 2x}$$

$$(1+ y+ 2x)dy= (1- 2y- 4x)dx$$

$$(1+ y+ 2x)dy+ (4x+ 2y- 1)dx= 0$$

Now we see that $(1+ y+ 2x)_x= 2= (4x+ 2y- 1)_y$ so this is an "exact differential"- there exist a function F(x,y) such that $dF= F_xdx+ F_ydy= (4x+2y- 1)dx+ (1+ y+ 2x)dx$.

From $F_x= 4x+ 2y- 1$ so that $F= 2x^2+ 2xy- g(y)$.
(Since F_x is a differentiation with respect to x, treating y as a constant, when we "reverse" that the "constant of integration" may be a function of y.)

From that, $F_y= 2x- g'(y)= 1+ y+ 2x$ so that $-g'(y)= 1+ y$ and $g'(y)= -1- y$. $g(y)= -y- \frac{1}{2}y^2+ C$.

So $F(x,y)= 2x^2+ 2xy- (-y- \frac{1}{2}y^2+ C)= 2x^2+ 2xy+ y+ \frac{1}{2}y^2+ c$.

Saying that "dF= 0" gives $F(x, y)= 2x^2+ 2xy- (-y- \frac{1}{2}y^2+ C)= 2x^2+ 2xy+ y+ \frac{1}{2}y^2+ c$ equals a constant. Including the "c" in that constant,
$2x^2+ 2xy+ y+ \frac{1}{2}y^2= C$.

 

[wozzers]

thank you all for your help i finally worked it out, i have a better understanding using the substitution method hallsofivy i still need to practice a few more problems to be confident in my ability to solve the type of DE using the Intial condition i got y= sqrt (6x+4) -2x-1 quite happy although the question is very tedious was required utilizing the completing the square technique to completely solve it if anyone knows where i can find more problems like this with solutions please assist

 

[Ackbach]

...if anyone knows where i can find more problems like this with solutions please assist

The ODE's book I mentioned has loads of these problems. You could also make up your own. I haven't tried Mathematica recently to see if it can solve these. WolframAlpha can't, and Sage (Maxima) can't.

 

[wozzers]

The ODE's book I mentioned has loads of these problems. You could also make up your own. I haven't tried Mathematica recently to see if it can solve these. WolframAlpha can't, and Sage (Maxima) can't.

i've looked in the book i am struggling to find these type of problems in the book do you know where exactly in the book these problems are?

 

[Ackbach]

The section in the book dealing with these kinds of equations is Lesson 8, starting on Page 62. The exercises are on page 69, with answers immediately following. Incidentally, this entire section of the book is available on Google Books.

 

[Bill Simpson]

I haven't tried Mathematica recently to see if it can solve these. WolframAlpha can't

WolframAlpha can solve this:

http://www.wolframalpha.com/input/?i=DSolve[y'[x]+==+(1+-+2+y[x]+-+4+x)/(1+++y[x]+++2+x),+y[x],+x]

Mathematica does the same.

Inside the radical simplifies to 1+6x+c

and checking by subsituting and differentiating seems to verify the result.

 

[Ackbach]

WolframAlpha can solve this:

http://www.wolframalpha.com/input/?i=DSolve[y'[x]+==+(1+-+2+y[x]+-+4+x)/(1+++y[x]+++2+x),+y[x],+x]

Mathematica does the same.

Inside the radical simplifies to 1+6x+c

and checking by subsituting and differentiating seems to verify the result.

Good catch. I think I was making the mistake of putting in y, instead of the more explicit y[x].

 

[epenguin]

i am having issues solving an ODE it is given as

y'= (1-2y-4x)/(1+y+2x)

I've been told to find the particular solution when y(0)=1

please help

This looks to me like just a warming up excercise preparatory to treating the more general case called the 1st order linear homogeneous equation I think. They are hoping you'd notice that the RHS is a function of (y + 2x). Call this Y, ask yourself (e.g.) what dY/dx is and it all works out nice and simple I think.

Edit. I see you did notice this, #3. Maybe another look?

 

[epenguin]

Since solutions have been given, with my version with the substitution I mentioned we have y' = Y' - 2 and get

$$Y' = \frac{3}{1+Y}$$

We can easily solve and incorporate the initital condition to give I think

y² + 4xy + 4x² + 2y - 2x - 3 = 0

And we could leave it like that.



But if it is wanted factorised, snce it was mentioned doing this is a bore, it appears easier to start from the form in which this is first solved:

Y² + 2Y = 6x + 3

(initial condition incorporated here). Write this

Y² + 2Y +1 - 6x - 4 = 0

then proceed

(Y + 1)² = 6x + 4

(y + 2x + 1)² = 6x + 4

(y + 2x + 1) = ± √(6x + 4)

y = -(2x + 1) ± √(6x + 4)

which appears the same as answers given.

 

[Ray Vickson]

The ODE's book I mentioned has loads of these problems. You could also make up your own. I haven't tried Mathematica recently to see if it can solve these. WolframAlpha can't, and Sage (Maxima) can't.

Maple finds a solution with no trouble.