Question

Y×x×y'=(y^2+9)×(x^2+4)

Answer 1

$yxy'=(y^2+9)(x^2+4)\\\\\frac{dy}{dx}=\frac{(y^2+9)(x^2+4)}{yx}\\\\\int \frac{y\, dy}{y^2+9}=\int \frac{(x^2+4)\, dx}{x}\\\\\frac{1}{2}\int \frac{2y\, dy}{y^2+9}=\int (x+\frac{4}{x})\, dx\\\\\frac{1}{2}\cdot ln(y^2+9)=\frac{x^2}{2}+4\, ln|x|+C$

Answer 2

The equation you provided is:

Y × x × y' = (y^2 + 9) × (x^2 + 4)

This appears to be a first-order nonlinear ordinary differential equation. To solve it, we can use separation of variables and integrate both sides.

Let's rewrite the equation in a slightly different form for clarity:

y' = (y^2 + 9) × (x^2 + 4) / (Y × x)

To solve the equation, we'll follow these steps:

Step 1: Separate the variables. (y^2 + 9) / y' = (x^2 + 4) / (Y × x)

Step 2: Integrate both sides with respect to their respective variables. ∫ (y^2 + 9) / y' dy = ∫ (x^2 + 4) / (Y × x) dx

Step 3: Solve the integrals on each side. ∫ (y^2 + 9) / y' dy = ∫ (x^2 + 4) / (Y × x) dx

The left-hand side integral is a bit tricky, so we'll use a substitution to simplify it. Let's substitute u = y^2 + 9, which gives us du = 2y dy.

∫ (1/2) du / y' = ∫ (x^2 + 4) / (Y × x) dx

Step 4: Simplify and integrate further. (1/2) ln|u| = ∫ (x^2 + 4) / (Y × x) dx

Step 5: Substitute u back in terms of y. (1/2) ln|(y^2 + 9)| = ∫ (x^2 + 4) / (Y × x) dx

Step 6: Solve the right-hand side integral. To evaluate the integral, we can divide the integrand into two fractions: ∫ (x^2 + 4) / (Y × x) dx = ∫ (x^2 / (Y × x) + 4 / (Y × x)) dx = (1/Y) ∫ (x dx) + (4/Y) ∫ (1/x) dx = (1/Y) ∫ (x dx) + (4/Y) ln|x| + C

Where C is the constant of integration.

Step 7: Substitute the result back into the equation. (1/2) ln|(y^2 + 9)| = (1/Y) ∫ (x dx) + (4/Y) ln|x| + C

Step 8: Simplify further and solve for y. ln|(y^2 + 9)|^(1/2) = (1/Y) (x^2/2) + (4/Y) ln|x| + C

|y^2 + 9|^(1/2) = e^[(1/Y) (x^2/2) + (4/Y) ln|x| + C]

Now we can solve for y by taking the square root of both sides and considering both the positive and negative square root.

y^2 + 9 = ± e^[(1/Y) (x^2/2) + (4/Y) ln|x| + C]

Finally, we can solve for y by taking the square root again:

y = ± sqrt(± e^[(1/Y) (x^2/2) + (4/Y) ln|x| + C] - 9)

Note that there are multiple possible solutions due to the presence of absolute values and the constant