Question #308138

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)



1
Expert's answer
2022-03-14T17:36:43-0400

a)..

To solve the separable equation dy(x)dx=116xy(x):Divide both sides by y(x)16:\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} :\\

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

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


16dy(x)dxy(x)dx=xdx\int \frac{16 \frac{d y(x)}{d x}}{y(x)} d x=\int x d x \\[4mm]

Evaluate the integrals:16log(y(x))=x22+c1, where c1 is an arbitrary constant.\text{Evaluate the integrals:}\\ 16 \log (y(x))=\frac{x^{2}}{2}+c_{1}, \text{ where } c_{1} \text{ is an arbitrary constant.}


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


y(x)=e1/32(x2+2c1)y(x)=e^{1 / 32\left(x^{2}+2 c_{1}\right)}

Simplifying the arbitrary constants, we have the implicit function to be:y(x)=e1/32(x2+c1)\text{Simplifying the arbitrary constants, we have the implicit function to be:}\\ y(x)=e^{1 / 32\left(x^{2}+c_{1}\right)}

b).


Solve for c1using the initial conditions:Substitute y(4)=1 into y(x)=e1/32(x2+c1):e1/32(c1+16)=1Solving the equation:c1=16Substitute c1=16 into y(x)=e1/32(x2+c1):Answer:y(x)=e1/32(x216)\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).

Solve for c1 using the initial conditions:Substitute y(0)=2 into y(x)=e1/32(x2+c1):ec1/32=2Solve the equation:c1=32iπ+32log(2)Substitute c1=32iπ+32log(2) into y(x)=e1/32(x2+c1):Answer:y(x)=2ex2/32\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}


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