If y is the solution of the differential equation , y(0)=1, the value of y(-1) is

The correct answer is 0.

Step 1: Identify the Differential Equation from the Image
As shown in the image, the differential equation to be solved is:
$$y^3\frac{dy}{dx}+x^3=0$$
along with the initial condition $y\left(0\right)=1$. We are required to find the value of $y(-1)$.

Step 2: Separation of Variables
We can rewrite the equation by isolating the terms involving $y$ on one side and the terms involving $x$ on the other side:
$$y^3\frac{dy}{dx}=-x^3$$
Multiplying both sides by $dx$, we separate the variables:
$$y^{3}dy=-x^{3}dx$$

Step 3: Integrate Both Sides
Integrating both sides of the separated equation:
$$\int y^{3}dy=-\int x^{3}dx+C$$
where $C$ is the constant of integration.
Evaluating the integrals using the power rule:
$$\frac{y^4}{4}=-\frac{x^4}{4}+C$$
Rearranging the terms gives the general solution:
$$\frac{x^4}{4}+\frac{y^4}{4}=C$$

Step 4: Determine the Constant C using the Initial Condition
We are given $y\left(0\right)=1$, meaning $y=1$ when $x=0$. Substituting these coordinates into the general equation:
$$\frac{0^4}{4}+\frac{1^4}{4}=C$$
$$C=\frac{1}{4}$$
Therefore, the particular solution is:
$$\frac{x^4}{4}+\frac{y^4}{4}=\frac{1}{4}$$
Multiplying by 4 simplifies it to:
$$x^4+y^4=1$$

Step 5: Compute y(-1)
To find the value of $y(-1)$, we substitute $x=-1$ into our particular equation:
$${(-1)}^4+y^4=1$$
Since ${(-1)}^4=1$, this simplifies to:
$$1+y^4=1$$
$$y^4=0$$
$$y=0$$
Thus, $y(-1)=0$.