I am solving problems found in the miscellaneous section of Chapter 2 that involve second order linear equations that can be transformed into first-order. Second-order isn't introduced until Chapter 3.
Here's what's given:
Equations with the Independent Variable Missing. Consider second order differential equations of the form y''=f(y,y'),in which the independent variable t does not appear explicitly. If we let v=y', then we obtain dv/dt=f(y,v). Since the right side of this equation depends on y and v,rather than on t and v,this equation contains too many variables. However, if we think of y as the independent variable, then by the chain rule, dv/dt =(dv/dy)(dy/dt)=v(dv/dy). Hence the original differential equation can be written as v(dv/dy)=f(y,v). Provided that this first order equation can be solved, we obtain v as a function of y. A relation between y and t results from solving dy/dt=v(y), which is a separable equation. Again, there are two arbitrary constants in the final result ($c_1$ and $c_2$).
1) $yy''+(y')^2 = 0$
2) $2y^2y''+2y(y')^2 =1$
Thanks. I've asked a similar question on solving these problems where dependent variable is unknown, and I was able to solve them all without too much difficulty. These problems however I just don't know where to begin. If someone could show me how to get to the integration part at least, I should be able to handle from there.
Update: Here's a screenshot for solution of problem 1. I think I've made a mistake in calculation somewhere.
Best Answer
For the first OE, set $$y=e^{\int u(x)dx}$$ So $$y'=ue^{\int udx}$$ and $$y''=(u'+u^2)e^{\int udx}$$ Now we have $yy''+(y')^2=(u'+u^2)e^{2\int udx}+u^2e^{2\int udx}$ which is zero. Since $e^{2\int udx}\neq 0$ then you should solve $u'+2u^2=0$.