# Find Sol for ODE dy/dx=(x+y+2)^2

1. Sep 16, 2013

### 1s1

1. The problem statement, all variables and given/known data
Find the general solution:
$$\frac{dy}{dx}=(x+y+3)^{2}$$

2. Relevant equations

3. The attempt at a solution
Methods I have learned: separation of variables, integrating factor for linear equations, exact equations, and substitution. I don't even know where to begin on this one. It has a $y^{2}$ term so it isn't linear; it isn't an exact equation; so there must be a substution. But it doesn't look anything like the substitution problems I've done. I expanded it, but that just seems to make it really long. Any hints or help would be greatly appreciated!

2. Sep 16, 2013

### fzero

Obviously the difficulty is that we have $x+y$ appearing in the power on the RHS, so an obvious choice of substitution would involve this combination. Try $u=x+y+3$. Since this is linear, $dy/dx$ and $du/dx$ are simply related.

3. Sep 16, 2013

### 1s1

Thanks! Clearly, I don't understand how substitution works ... yet.

The only substitution we've learned so far is for Bernoulli's Equation, $\frac{dy}{dx}+P(x)y=f(x)y^{n}$

Where the substitution is $u=y^{1-n}$

I'll run $u=x+y+3$ through and I should get a linear equation like you said.

Sorry for the newbie ODE question.

4. Sep 16, 2013

### fzero

Very often substitution involves trial and error, using experience to suggest promising approaches.

After the substitution the equation is still not linear (it involves $u^$), but it is separable, which allows us to easily solve it.

5. Sep 16, 2013

### 1s1

Making the suggested substitution: $u=x+y+3$ and using:
$$\int \frac{1}{a^{2}+u^{2}}du=\frac{1}{a}tan^{-1}\frac{u}{a}+C$$
$$x=tan^{-1}(x+y+3)+C$$

The choice of substitution in this case seems similar to what you would choose when doing integration by substitution, so hopefully that will be a trend and will make picking the stubstution easier. I suppose practicing with different types of substitutions also develops experience on likely beneficial substitutions. It also seems appropriate to choose $u$ so that $x$ and $y$ are simply related. Thanks again fzero!