Find the particular solution of the given differential equation
dydx=−4xe(y−x^2);y=11whenx=1.
THanks
sarah, I am not certain what you meant to type.
i get this
dy/dx= -4x * e^ what is the exponent of e?
the exponent is (y-x^2)
dy/dx=−4xe^(y−x^2)
dy/dx = -4xe^y e^(-x^2)
-e^-y dy = 4x e^(-x^2)
e^-y = c-2e^(-x^2)
y = -ln(c - 2e^(-x^2))
since y(1) = 11,
-ln(c-2e^-1) = 11
c - 2e^-1 = e^-11
c = 2e^-1 + e^-11
To find the particular solution of the given differential equation, we can use the method of separation of variables. Here's the step-by-step process:
1. Start with the given differential equation: dy/dx = -4xe(y - x^2).
2. Rearrange the equation by dividing both sides by (y - x^2):
(1/(y - x^2)) dy/dx = -4x.
3. Now, we can separate the variables by multiplying both sides by dx and dividing both sides by (y - x^2):
(1/(y - x^2)) dy = -4x dx.
4. Integrate both sides with respect to their respective variables:
∫(1/(y - x^2)) dy = ∫(-4x) dx.
5. Evaluate the integrals:
ln|y - x^2| = -2x^2 + C, where C is the constant of integration.
6. Solve for y:
ln|y - x^2| = -2x^2 + C.
Take the exponential of both sides:
|y - x^2| = e^(-2x^2 + C).
7. Since e^C is a positive constant, we can remove the absolute value and write:
y - x^2 = Ce^(-2x^2).
8. Now, use the given initial condition y = 11 when x = 1 to find the value of the constant C:
11 - 1^2 = Ce^(-2(1)^2).
10 = C * e^(-2).
9. Solve for C:
C = 10 / e^(-2).
Using a calculator or approximation, we find that C ≈ 135.34.
10. Substitute the value of C back into the equation:
y - x^2 = (135.34)e^(-2x^2).
And there you have it! The particular solution to the given differential equation is y - x^2 = (135.34)e^(-2x^2).