Let's solve the initial value problem step by step: Given: \[ y' = 3t y^2, \quad y(0) = y_0 \] --- ### **Step 1: Separate the variables** \[ \frac{dy}{dt} = 3t y^2 \] Divide both sides by \( y^2 \): \[ \frac{1}{y^2} \frac{dy}{dt} = 3t \] \[ \frac{dy}{y^2} = 3t\,dt \] --- ### **Step 2: Integrate both sides** Integrate both sides: \[ \int \frac{dy}{y^2} = \int 3t\,dt \] \[ \int y^{-2} dy = 3 \int t dt \] \[ - y^{-1} = \frac{3}{2} t^2 + C \] --- ### **Step 3: Solve for \( y \)** \[ - \frac{1}{y} = \frac{3}{2} t^2 + C \] \[ \frac{1}{y} = -\left( \frac{3}{2} t^2 + C \right) \] \[ y = \frac{1}{ -\left( \frac{3}{2} t^2 + C \right) } \] --- ### **Step 4: Use the initial condition to solve for \( C \)** Given \( y(0) = y_0 \): \[ y(0) = \frac{1}{ -C } = y_0 \] \[ -C = \frac{1}{y_0} \] \[ C = -\frac{1}{y_0} \] --- ### **Step 5: Write the explicit solution** Plugging \( C \) back: \[ y = \frac{1}{ -\left( \frac{3}{2} t^2 - \frac{1}{y_0} \right) } \] \[ y = \frac{1}{ -\frac{3}{2} t^2 + \frac{1}{y_0} } \] Or, combine numerators: \[ y = \frac{1}{ \frac{1}{y_0} - \frac{3}{2} t^2 } \] --- ### **Step 6: Interval of existence** The solution exists as long as the denominator is **not zero**: \[ \frac{1}{y_0} - \frac{3}{2} t^2 \neq 0 \] \[ \frac{3}{2} t^2 \neq \frac{1}{y_0} \] \[ t^2 \neq \frac{2}{3 y_0} \] Thus, the solution exists for: \[ |t| < \sqrt{\frac{2}{3 y_0}} \quad \text{if} \quad y_0 > 0 \] If \( y_0 < 0 \), then \( \frac{2}{3y_0} \) is negative, so there is **no real \( t \)** where the denominator is zero, so the solution exists for all \( t \). --- ## **Final Answer:** ### The solution is: \[ \boxed{ y(t) = \frac{1}{\frac{1}{y_0} - \frac{3}{2}t^2} } \] ### The interval of existence depends on \( y_0 \) as follows: - **If \( y_0 > 0 \):** \[ |t| < \sqrt{\frac{2}{3y_0}} \] - **If \( y_0 < 0 \):** The solution exists for all real \( t \). - **If \( y_0 = 0 \):** The solution is undefined for all \( t \) (division by zero in the denominator).

The explicit solution is: \[ \boxed{ y(t) = \frac{1}{\frac{1}{y_0} - \frac{3}{2} t^2} } \] **Interval of existence:** - For \( y_0 > 0 \): \[ |t| < \sqrt{\frac{2}{3 y_0}} \] - For \( y_0 < 0 \): \[ \text{Solution exists for all real } t \] - For \( y_0 = 0 \): \[ \text{Solution is undefined} \quad (\text{no solution}) \]