dy/dx = 4ye^(5x)
a) Separate the differential equation, then integrate both sides.
b) Write the general solution as a function y(x).
For the second part, I got y(x)=e^((5e^(5x))/(5)) + C but I don't understand how to separate differential equations and/or integrate both sides...
dy / y = 4 e^(5x) dx
ln y = (4/5) e^(5x) + c
e^ln y = y = e^ [(4/5)e^(5x) +c)
y = e^c e^[(4/5)e^(5x) ]
y = C e^[(4/5)e^(5x) ]
To solve the differential equation dy/dx = 4ye^(5x), you can follow these steps:
a) Separate the differential equation:
The first step is to separate the variables, which means isolating the y terms on one side of the equation and the x terms on the other side.
Start with the given equation: dy/dx = 4ye^(5x)
Divide both sides by "y" to get: (1/y)dy = 4e^(5x)dx
b) Integrate both sides:
Now that the equation is separated, you can integrate both sides with respect to their respective variables.
Integrate the left side: ∫(1/y)dy = ∫dy/y = ln|y| + C1, where C1 is the constant of integration.
Integrate the right side: ∫4e^(5x)dx = (4/5)e^(5x) + C2, where C2 is the constant of integration.
So the equation becomes: ln|y| + C1 = (4/5)e^(5x) + C2
c) Combine the constants:
Combine the constants of integration into a single constant, C = C1 - C2, where C is the new constant of integration.
The equation becomes: ln|y| = (4/5)e^(5x) + C
d) Solve for y(x):
To solve for y(x), you need to eliminate the natural logarithm. Take the exponent of both sides:
e^(ln|y|) = e^((4/5)e^(5x) + C)
The natural logarithm and the exponential function cancel each other out:
|y| = e^((4/5)e^(5x) + C)
Remove the absolute value:
y = ± e^((4/5)e^(5x) + C)
e) Simplify the expression:
Now, simplify the expression by combining the constants:
y = e^(4e^(5x)/5 + C)
Since "e^C" represents an arbitrary constant, we can rewrite it as y = e^(4e^(5x)/5) * e^C = Ke^(4e^(5x)/5), where K is the new constant.
Therefore, the general solution is y(x) = Ke^(4e^(5x)/5).