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The solution of the differential equation (dy/dx) = ky, y(0) = c is
- a)x = ce-ky
- b)x = kecy
- c)y = cekx
- d)y = ce-kx
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
The solution of the differential equation (dy/dx) = ky, y(0) = c isa)x...
The given differential equation is, (dy/dx) = ky
dy/y = kdx
On integrating both the sides, we get
ln y = kx + ln c
y = cekx
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The solution of the differential equation (dy/dx) = ky, y(0) = c isa)x...
Solution:
The given differential equation is
(dy/dx) = ky
To solve this differential equation, we can separate the variables by moving all terms involving y to one side and all terms involving x to the other side:
(dy/y) = k(dx)
Next, we can integrate both sides of the equation. The integral of (dy/y) is ln|y|, and the integral of k(dx) is kx + C, where C is the constant of integration:
ln|y| = kx + C
To eliminate the absolute value, we can exponentiate both sides of the equation:
e^(ln|y|) = e^(kx + C)
This simplifies to:
|y| = e^(kx) * e^C
Since e^C is a constant, we can replace it with another constant, let's say A:
|y| = Ae^(kx)
Now, we can consider two cases for the absolute value:
Case 1: y > 0
In this case, the absolute value can be removed:
y = Ae^(kx)
Case 2: y < />
In this case, the absolute value becomes a negative sign:
-y = Ae^(kx)
Multiplying both sides by -1, we get:
y = -Ae^(kx)
Combining both cases, we can write the solution as:
y = Ce^(kx)
where C = A if y > 0, and C = -A if y < />
Given that y(0) = c, we can substitute this initial condition into the solution:
c = Ce^(k*0)
Since e^0 = 1, this simplifies to:
c = C
Therefore, the constant C is equal to c. Substituting this back into the solution, we get:
y = ce^(kx)
Hence, the correct solution to the given differential equation is option 'C': y = ce^(kx).
The given differential equation is
(dy/dx) = ky
To solve this differential equation, we can separate the variables by moving all terms involving y to one side and all terms involving x to the other side:
(dy/y) = k(dx)
Next, we can integrate both sides of the equation. The integral of (dy/y) is ln|y|, and the integral of k(dx) is kx + C, where C is the constant of integration:
ln|y| = kx + C
To eliminate the absolute value, we can exponentiate both sides of the equation:
e^(ln|y|) = e^(kx + C)
This simplifies to:
|y| = e^(kx) * e^C
Since e^C is a constant, we can replace it with another constant, let's say A:
|y| = Ae^(kx)
Now, we can consider two cases for the absolute value:
Case 1: y > 0
In this case, the absolute value can be removed:
y = Ae^(kx)
Case 2: y < />
In this case, the absolute value becomes a negative sign:
-y = Ae^(kx)
Multiplying both sides by -1, we get:
y = -Ae^(kx)
Combining both cases, we can write the solution as:
y = Ce^(kx)
where C = A if y > 0, and C = -A if y < />
Given that y(0) = c, we can substitute this initial condition into the solution:
c = Ce^(k*0)
Since e^0 = 1, this simplifies to:
c = C
Therefore, the constant C is equal to c. Substituting this back into the solution, we get:
y = ce^(kx)
Hence, the correct solution to the given differential equation is option 'C': y = ce^(kx).
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The solution of the differential equation (dy/dx) = ky, y(0) = c isa)x = ce-kyb)x = kecyc)y = cekxd)y = ce-kxCorrect answer is option 'C'. Can you explain this answer?
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