find an equation of the curve that satisfies dy/dx=4x^3y and whose y-intercept is 7
Why did the curve go to therapy? Because it had trouble finding its y-intercept!
To find the equation of the curve that satisfies dy/dx = 4x^3y and has a y-intercept of 7, we can separate the variables and integrate both sides.
Starting with dy/dx = 4x^3y, we rearrange it as dy/y = 4x^3dx.
Integrating both sides:
∫(1/y)dy = ∫4x^3dx.
ln|y| = x^4 + C,
where C is the constant of integration.
Since the curve passes through the point (0, 7), we can substitute these values to find C:
ln|7| = 0^4 + C,
ln|7| = C.
Therefore, the equation of the curve is:
ln|y| = x^4 + ln|7|.
Done! Now you've got a curve that's all set for some therapy sessions.
To find the equation of the curve that satisfies the given conditions, we can solve the differential equation and use the point (0, 7) as the initial condition.
The given differential equation is dy/dx = 4x^3y.
Let's solve this differential equation:
Separate the variables:
(dy/y) = 4x^3*dx
Integrate both sides:
∫(dy/y) = ∫(4x^3*dx)
ln|y| = x^4 + C1
Now we can exponentiate both sides:
|y| = e^(x^4 + C1)
Since we are dealing with a y-intercept of 7, we can rewrite the equation as:
y = ±e^(x^4 + C)
We need to find the appropriate constant, C, by substituting the y-intercept coordinates (0,7) in the equation:
7 = ±e^(0^4 + C)
7 = ±e^(0 + C)
7 = ±e^C
Taking the natural logarithms of both sides:
ln(7) = ln(±e^C)
ln(7) = C
The constant, C, is ln(7).
Therefore, the equation of the curve is:
y = ±e^(x^4 + ln(7))
Simplifying the equation, we get:
y = ±e^(x^4) * e^(ln(7))
Since e^(ln(7)) = 7, the final equation of the curve is:
y = ±7e^(x^4)
Note: The ± symbol represents that we have two curves, one reflecting over the x-axis of the other.
To find an equation of the curve that satisfies the given condition, we can use separation of variables. Here's how you can solve it step by step:
1. Start with the given differential equation: dy/dx = 4x^3y.
2. Rearrange the equation to separate the variables: dy/y = 4x^3dx.
3. Integrate both sides of the equation:
∫(dy/y) = ∫(4x^3 dx).
4. Solve the integrals:
ln|y| = x^4 + C, where C is the constant of integration.
5. Exponentiate both sides of the equation to eliminate the natural logarithm:
e^(ln|y|) = e^(x^4 + C).
Simplifying, we have:
|y| = e^(x^4) * e^C.
6. Since y-intercept is the point where x = 0, substitute the coordinates (0, 7) into the equation.
When x = 0, the right side of the equation becomes e^C.
Therefore, |7| = e^C.
Since the value inside the absolute value bars is positive, we can get rid of the absolute value sign:
7 = e^C.
7. Substituting this value for C back into the equation, we get:
|y| = e^(x^4) * 7.
8. Finally, we can drop the absolute value sign since e^(x^4) * 7 will always be positive, yielding the equation:
y = e^(x^4) * 7.
So, the equation of the curve that satisfies the given conditions and whose y-intercept is 7 is y = e^(x^4) * 7.
dy/y=4x^3 dx
lny=x^4 + C
when x=0, y=7, solve for C
I don't understand what is going on here, you keep posting these very simple problems. Surely you are not stuck on them.