Let's solve the problem step by step as requested: --- ### **Step 1: Check for Exactness** Given the differential equation: \[(x^2 \sin x + 4y)\,dx + x\,dy = 0,\] or, \[(x^2 \sin x + 4y)\,dx + (x)\,dy = 0.\] Let \( M(x,y) = x^2 \sin x + 4y \) and \( N(x,y) = x \). A differential equation \( M(x, y)\,dx + N(x, y)\,dy = 0 \) is **exact** if: \[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \] Let's compute these derivatives: - \(\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(x^2 \sin x + 4y) = 4\) - \(\frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(x) = 1\) Since \( 4 \neq 1 \), the equation is **not exact**. **Explanation:** The equation is not exact because the mixed partial derivatives are not equal. --- ### **Step 2: Find an Integrating Factor and Solve** Since the equation is not exact, let's try to find an integrating factor. Here, since \( \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} = \frac{4-1}{x} = \frac{3}{x} \), which is a function of \(x\) only. So, the integrating factor is: \[ \mu(x) = e^{\int \frac{3}{x} dx} = e^{3\ln|x|} = |x|^3 \] (For \(x > 0\), we can write this as \(x^3\).) Multiply both \(M\) and \(N\) by \(x^3\): \[ M^* = x^3(x^2 \sin x + 4y) = x^5 \sin x + 4x^3 y \] \[ N^* = x^3 \cdot x = x^4 \] Now check for exactness: \[ \frac{\partial M^*}{\partial y} = 4x^3 \] \[ \frac{\partial N^*}{\partial x} = 4x^3 \] Now, the equation is exact. **Explanation:** By multiplying by \(x^3\), we made the equation exact. --- #### **Now Solve the Exact Equation** We have: \[ (x^5 \sin x + 4x^3 y)\,dx + x^4\,dy = 0 \] Let \(F(x, y)\) be a potential function such that: \[ \frac{\partial F}{\partial x} = M^* = x^5 \sin x + 4x^3 y \] \[ \frac{\partial F}{\partial y} = N^* = x^4 \] Integrate \(M^*\) with respect to \(x\): \[ F(x, y) = \int (x^5 \sin x + 4x^3 y) dx \] \[ = \int x^5 \sin x dx + 4y \int x^3 dx \] \[ = \int x^5 \sin x dx + y \cdot x^4 + h(y) \] Let’s compute \( \int x^5 \sin x dx \) (use tabular integration by parts or reduction formula, but for conciseness, let's denote this as \(S(x)\)): Let’s denote: \[ S(x) = \int x^5 \sin x dx \] So, \[ F(x, y) = S(x) + x^4 y + C \] Now, take derivative with respect to \(y\): \[ \frac{\partial F}{\partial y} = x^4 \] which matches \(N^*\), so potential function is correct. --- ### **Final Implicit Solution** The general solution is: \[ S(x) + x^4 y = C \] Apply the initial condition \(y(0) = 1\): First, \(x = 0\), \(y = 1\): \[ S(0) + 0^4 \cdot 1 = C \implies S(0) = C \] But \[ S(0) = \int_0^0 x^5 \sin x dx = 0 \] So \(C = 0\). #### **Final Answer (Implicit solution):** \[ \boxed{ \int x^5 \sin x dx + x^4 y = 0 } \] Or, \[ \boxed{ S(x) + x^4 y = 0, \quad y(0) = 1 } \] If you want the explicit form, solve for \(y\): \[ y = -\frac{1}{x^4}\int x^5 \sin x dx \] But the main answer is the implicit form above. --- **Summary of Steps:** 1. **Checked for exactness:** Not exact. 2. **Found integrating factor \(x^3\), made exact, and solved using potential function.** Let me know if you want the explicit form of \( \int x^5 \sin x dx \) (it involves repeated integration by parts)!