What is the general soln of dy/dx=9(x^2)(y) and particular soln of dy/dx=3x^2(e^-y)
dy/dx = 9x^2 y
dy/y = 9x^2 dx
ln y = 3x^3 + c
y = c e^(3x^3)
dy/dx=3x^2(e^-y)
e^y dy = 3x^2 dx
e^y = x^3 + c
y = ln(x^3+c)
Can't provide a particular solution unless you can tell me something else about y or c.
To find the general solution and particular solution of a differential equation, we need to solve the equation by integrating both sides with respect to x.
Let's start with the first equation: dy/dx = 9(x^2)(y).
Step 1: Separate variables by moving all terms involving y to one side and terms involving x to the other side:
dy/y = 9(x^2) dx.
Step 2: Integrate both sides:
∫(1/y)dy = ∫9(x^2)dx.
The integral of 1/y with respect to y is ln|y| + C1, where C1 is a constant of integration.
The integral of 9(x^2) with respect to x is 3x^3 + C2, where C2 is another constant of integration.
Step 3: Combine the integrals and constants:
ln|y| + C1 = 3x^3 + C2.
Step 4: Simplify the equation by combining the constants:
ln|y| = 3x^3 + (C2 - C1).
Step 5: Eliminate the absolute value by taking the exponential of both sides:
e^(ln|y|) = e^(3x^3 + (C2 - C1)).
The absolute value becomes:
|y| = e^(3x^3 + (C2 - C1)).
Step 6: Remove the absolute value by considering two separate cases:
For y > 0:
y = e^(3x^3 + (C2 - C1)).
For y < 0:
y = -e^(3x^3 + (C2 - C1)).
These equations represent the general solution to the differential equation dy/dx = 9(x^2)(y).
Now let's move on to the second equation: dy/dx = 3x^2(e^(-y)).
Step 1: Separate variables:
dy/e^(-y) = 3x^2 dx.
Step 2: Integrate both sides:
∫e^y dy = ∫3x^2 dx.
The integral of e^y with respect to y is e^y + C3, where C3 is a constant of integration.
The integral of 3x^2 with respect to x is x^3 + C4, where C4 is another constant of integration.
Step 3: Combine the integrals and constants:
e^y + C3 = x^3 + C4.
Step 4: Simplify the equation by combining the constants:
e^y = x^3 + (C4 - C3).
Step 5: Solve for y by taking the natural logarithm of both sides:
ln(e^y) = ln(x^3 + (C4 - C3)).
y = ln(x^3 + (C4 - C3)).
This equation represents the particular solution to the differential equation dy/dx = 3x^2(e^(-y)).