4.93. Solve the Cauchy problem with the differential equation \(3 y^{\prime \prime}=4 \frac{x^{3}}{\left(y^{\prime}\right)^{2}}\) and the initial condition \(y(1)=0, y^{\prime}(1)=2\). After the substitution \(y^{\prime}=p(x)\), we obtain the equation \(3 p^{\prime}=4 \frac{x^{3}}{p^{2}}\).
Solution. Let's find its general solution, for this we rewrite the equation in the form: $3 \frac{d p}{d x}=4 \frac{x^{3}}{p^{2}}$. We separate the variables: $3 p^{2} d p=4 x^{3} d x$. We integrate both sides with respect to their variables:

$$
\int 3 p^{2} d p=\int 4 x^{3} d x, 3 \frac{p^{3}}{3}=4 \frac{x^{4}}{4}+C, \quad p^{3}=x^{4}+C
$$

We have obtained the general solution in implicit form. Let's find the general solution in explicit form: $p=\sqrt[3]{x^{4}+C}$.

Using the initial condition, we find the specific numerical value of the arbitrary constant $C$ and write down the particular solution of the Cauchy problem:

$$
2=\sqrt[3]{1^{4}+C}, 2^{3}=1+C, \quad C=7, \quad p=\sqrt[3]{x^{4}+7}
$$

From here, taking into account $y(1)=0$

$$
y(x)=\int_{1}^{x} p(t) d t=\int_{1}^{x} \sqrt[3]{t^{4}+7} d t
$$

This integral does not have antiderivatives in terms of elementary functions. In subsequent parts of the manual, it will be shown how integrals of this type are found numerically.