MTH 256 Funk Writing Assignment 1) So far, some differential equations you have studied have involved dependent variables raised to the 1st power; i

MTH 256 Funk Writing Assignment
1) So far, some differential equations you have studied have involved dependent variables raised to the 1st power; i.e. linear equations. This problem is about a technique, attributed to Bernoulli, that involves solving a 1st order, nonlinear differential equation, like the following: ?? + 1 ?? = ?2?4 ; ?(1) = 1 ?? 
Substitute ? = ?−3 to convert this nonlinear equation into one that is linear in the variable ?. You will need to fix the derivative term using the chain 
?? rule; that is, 
?? 
?? = 
?? 
?? . You will then solve the resulting equation using ?? 
the integrating factor method, after some rewriting. This will give you the solution ? as a function of ?. Using the original substitution in reverse, you can get the explicit solution ? as a function of ?. ' Rn’t !RrJHs?R1IP SOP HM the given point to specify the arbitrary constant of your general solution (to get the specific solution for this problem). 
2. For this problem, we turn to 2nd order differential equations. In class, the focus is with those equations having constant coefficients; however, there are other 2nd order differential equations with more general coefficients, and in this problem, these coefficients will be special polynomial coefficients, as illustrated in the following 2nd order differential equation: 4?2?″ + 8??′ — 3? = 0 
Due to the polynomial coefficients, the natural guess for the form of solution is that of power functions; i.e. ? = ??. Substitute this form, along with its 1st and 2nd derivatives, into the differential equation; you will end up with another kind of characteristic equation involving the power variable ?. Use algebra to solve for ? (should be two values), then write the general solution using the superposition principle. Finally, determine the specific solution satisfying the conditions ?(1) = 14 and ?′(1) = 3.