Answer to Question #308138 in Differential Equations for Papi Chulo

Question

Solve the differential equation 

dy/dx=x/16y

.

a) Find the implicit solution

b) Find the equation of the solution through the point (x,y)=(4,1) Your equation must describe a single curve of y=f(x) with the domain of f as large as possible. 

c) Find the equation of the solution through the point (x,y)=(0,−2) Your answer should be of the form y=f(x)

Answer 1

a)..

$\text{To solve the separable equation } \frac{d y(x)}{d x}=\frac{1}{16} x y(x) :\\ \text{Divide both sides by } \frac{y(x)}{16} :\\$

$$\frac{16 \frac{d y(x)}{d x}}{y(x)}=x\\[2mm]$$

$\text{Integrate both sides with respect to }x :$

$$\int \frac{16 \frac{d y(x)}{d x}}{y(x)} d x=\int x d x \\[4mm]$$

$\text{Evaluate the integrals:}\\ 16 \log (y(x))=\frac{x^{2}}{2}+c_{1}, \text{ where } c_{1} \text{ is an arbitrary constant.}$

$\text{Solve for } y(x):$

$$y(x)=e^{1 / 32\left(x^{2}+2 c_{1}\right)}$$

$$\text{Simplifying the arbitrary constants, we have the implicit function to be:}\\ y(x)=e^{1 / 32\left(x^{2}+c_{1}\right)}$$

b).

$\text{Solve for }c_{1} \text{using the initial conditions:}\\ \text{Substitute } y(4)=1 \text{ into } y(x)=e^{1 / 32\left(x^{2}+c_{1}\right)} : e^{1 / 32\left(c_{1}+16\right)}=1\\[4mm] \text{Solving the equation:}\\ c_{1}=-16\\[4mm] \text{Substitute } c_{1}=-16 \text{ into } y(x)=e^{1 / 32\left(x^{2}+c_{1}\right)} :\\[4mm] \text{Answer:}\\ y(x)=e^{1 / 32\left(x^{2}-16\right)}$

c).

$\text{Solve for } c_{1} \text{ using the initial conditions:}\\ \text{Substitute } y(0)=-2 \text{ into } y(x)=e^{1 / 32\left(x^{2}+c_{1}\right)} : e^{c_{1} / 32}=-2\\[4mm] \text{Solve the equation:}\\ c_{1}=32 i \pi+32 \log (2)\\[4mm] \text{Substitute } c_{1}=32 i \pi+32 \log (2) \text{ into } y(x)=e^{1 / 32\left(x^{2}+c_{1}\right)} :\\[4mm] \text{Answer:}\\ y(x)=-2 e^{x^{2} / 32}$