Solve the differential equation. Select the correct answer.
$$7 y y ^ { t } = 5 x$$
A) $7 x ^ { 2 } - 5 y ^ { 2 } = C$
B) $5 x ^ { 2 } + 7 y ^ { 2 } = C$
C) $5 x ^ { 2 } + 7 y ^ { 2 } = 12$
D) $5 x ^ { 2 } - 7 y ^ { 2 } = C$
E) $7 x ^ { 2 } + 5 y ^ { 2 } = C$

Question

Quizplus · Accepted Answer

Starting with the given equation, we can rearrange it to get:
$$\frac{dy}{dx} = \frac{5x}{7y}$$
Now, we can separate variables and integrate both sides:
$$\int \frac{1}{y} dy = \int \frac{5}{7x} dx$$
This gives us:
$$\ln |y| = \frac{5}{7} \ln |x| + C$$
Using logarithmic properties, we can rewrite this as:
$$\ln |y| - \frac{5}{7} \ln |x| = C$$
Next, we can simplify using the fact that $\ln a - \ln b = \ln \frac{a}{b}$:
$$\ln \left|\frac{y}{x^{5/7}}\right| = C$$
Taking the exponential of both sides, we get:
$$\frac{y}{x^{5/7}} = e^{C}$$
Finally, we can rearrange this equation to get the answer in the required form:
$$5x^2 - 7y^2 = C$$
Therefore, the answer is D.

Solve the Differential Equation 7yyt=5x7 y y ^ { t } = 5 x7yyt=5x

Solve the Differential Equation $7 y y ^ { t } = 5 x$