Solve the given differential equation. $$ e^{x^{2}} d y=x \sqrt{3-y} d x $$ A. $$ 4 \sqrt{3-y}=e^{-x^{2}}+C $$ B. $$ 3 \sqrt{4-y}=e^{-x^{2}}+C $$ C. $$ \sqrt{3-y}=4 e^{-x^{2}}+C $$ D. $$ y=3-\frac{e^{-x^{4}}}{4}+C $$

Question

UpStudy AI · Accepted Answer

To solve the differential equation:

$$
e^{x^{2}} \, dy = x \sqrt{3 - y} \, dx
$$

we can follow these steps:

1. **Rewrite the Equation:**

Divide both sides by $$ e^{x^2} $$:

$$
   dy = x e^{-x^2} \sqrt{3 - y} \, dx
   $$

2. **Separate Variables:**

Bring all terms involving $$ y $$ to one side and $$ x $$ to the other:

$$
   \frac{dy}{\sqrt{3 - y}} = x e^{-x^2} \, dx
   $$

3. **Integrate Both Sides:**

- **Left Side:**

Let $$ u = 3 - y $$, then $$ du = -dy $$:

$$
     \int \frac{dy}{\sqrt{3 - y}} = -\int \frac{du}{\sqrt{u}} = -2\sqrt{u} = -2\sqrt{3 - y}
     $$

- **Right Side:**

Let $$ v = -x^2 $$, then $$ dv = -2x \, dx $$, so $$ x \, dx = -\frac{dv}{2} $$:

$$
     \int x e^{-x^2} \, dx = -\frac{1}{2} \int e^{v} \, dv = -\frac{1}{2} e^{-x^2} + C
     $$

4. **Combine the Results:**

$$
   -2\sqrt{3 - y} = -\frac{1}{2} e^{-x^2} + C
   $$

Simplify and solve for $$ \sqrt{3 - y} $$:

$$
   2\sqrt{3 - y} = \frac{1}{2} e^{-x^2} + C' \quad (\text{where } C' \text{ is a new constant})
   $$

Multiply both sides by 2:

$$
   4\sqrt{3 - y} = e^{-x^2} + C
   $$

This is the general solution to the differential equation.

**Final Answer:**

**A.** $$ 4 \sqrt{3-y}=e^{-x^{2}}+C $$
The solution to the differential equation is $$ 4 \sqrt{3 - y} = e^{-x^{2}} + C $$.

Solve the given differential equation.

A.
B.
C.
D.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions

Solve the given differential equation. A. B. C. D.