MHB Non-linear first order differential equation question

[talha1]

View attachment 1844

Kindly solve it and i need help to understand NON LINEAR Questions

 

[MarkFL]

Re: non linear first order differential equation question

Hello and welcome to MHB, talha! (Sun)

First, I need to say that our goal here at MHB is to help people solve their problems, not solve it for them. You will learn more through guided effort than simply being given the solution.

Second, I suspect the exponents on $e$ are squared as follows:

$$y'=\frac{2xye^{\left(\frac{x}{y} \right)^2}}{y^2+y^2e^{\left(\frac{x}{y} \right)^2}+2x^2e^{\left(\frac{x}{y} \right)^2}}$$

Is this correct?

 

[talha1]

Re: non linear first order differential equation question

yes it is correct

 

[MarkFL]

Re: non linear first order differential equation question

Okay, good. Now can you express the right side of the ODE as a function of $$\frac{x}{y}$$ only?

 

[MarkFL]

Re: non linear first order differential equation question

This is how I would work the problem:

Spoiler

We are given the ODE:

$$y'=\frac{2xye^{\left(\frac{x}{y} \right)^2}}{y^2+y^2e^{\left(\frac{x}{y} \right)^2}+2x^2e^{\left(\frac{x}{y} \right)^2}}$$

We observe that we must have $y\ne0$ so multiplying the right by $$1=\frac{y^{-2}}{y^{-2}}$$ will not result in the loss of any trivial solution:

$$y'=\frac{2\dfrac{x}{y}e^{\left(\frac{x}{y} \right)^2}}{1+e^{\left(\frac{x}{y} \right)^2}+2\left(\dfrac{x}{y} \right)^2e^{\left(\frac{x}{y} \right)^2}}$$

Now the ODE is in the form $$y'=f\left(\frac{x}{y} \right)$$ and we may use the substitution:

$$v=\frac{y}{x}\implies y=vx\,\therefore y'=v+xv'$$

Hence, the ODE may now be written:

$$v+xv'=\frac{\dfrac{2}{v}e^{v^{-2}}}{1+e^{v^{-2}}+\dfrac{2}{v^2}e^{v^{-2}}}$$

Multiplying the right side by $$1=\frac{v^2}{v^2}$$ we obtain:

$$v+xv'=\frac{2ve^{v^{-2}}}{v^2+v^2e^{v^{-2}}+2e^{v^{-2}}}$$

Subtracting through by $v$ and simplifying, we get:

$$xv'=-\frac{v^3\left(1+e^{v^{-2}} \right)}{v^2\left(1+e^{v^{-2}} \right)+2e^{v^{-2}}}$$

Separation of variables yields:

$$\frac{v^2\left(1+e^{v^{-2}} \right)+2e^{v^{-2}}}{v^3\left(1+e^{v^{-2}} \right)}\,dv=-\frac{1}{x}\,dx$$

On the left side, dividing each term by $$1+e^{v^{-2}}$$ allows us to write:

$$\left(\frac{1}{v}-\frac{-2v^{-3}e^{v^{-2}}}{1+e^{v^{-2}}} \right)\,dv=-\frac{1}{x}\,dx$$

Now observing that:

$$\frac{d}{dv}\left(1+e^{v^{-2}} \right)=-2v^{-3}e^{v^{-2}}$$

we may integrate as follows:

$$\ln|v|-\ln\left(1+e^{v^{-2}} \right)=-\ln|x|+C$$

Back-substituting for $v$, we obtain:

$$\ln\left|\frac{y}{x} \right|-\ln\left(1+e^{\left(\frac{x}{y} \right)^2} \right)=-\ln|x|+C$$

Adding $$\ln|x|$$ to both sides, we obtain:

$$\ln|y|-\ln\left(1+e^{\left(\frac{x}{y} \right)^2} \right)=C$$

Using a property of logs, we may rewrite the left side as:

$$\ln\left|\frac{y}{1+e^{\left(\frac{x}{y} \right)^2}} \right|=C$$

Converting from logarithmic to exponential form, we find:

$$\left|\frac{y}{1+e^{\left(\frac{x}{y} \right)^2}} \right|=e^{C}$$

If we redefine the constant on the right as any positive value (since $y\ne0$), we have:

$$\left|\frac{y}{1+e^{\left(\frac{x}{y} \right)^2}} \right|=C$$ where $$0<C$$

Now, if we redefine the constant again as any real value except zero, we may write:

$$\frac{y}{1+e^{\left(\frac{x}{y} \right)^2}}=C$$ where $$C\ne0$$

And if we so choose, we may arrange this as the implicit solution:

$$y=C\left(1+e^{\left(\frac{x}{y} \right)^2} \right)$$