a)..
To solve the separable equation dxdy(x)=161xy(x):Divide both sides by 16y(x):
y(x)16dxdy(x)=x Integrate both sides with respect to x:
∫y(x)16dxdy(x)dx=∫xdx Evaluate the integrals:16log(y(x))=2x2+c1, where c1 is an arbitrary constant.
Solve for y(x):
y(x)=e1/32(x2+2c1)
Simplifying the arbitrary constants, we have the implicit function to be:y(x)=e1/32(x2+c1)
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(x2−16)
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
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