Topic: Engineering Mathematics
Solution for the following differential equation with boundary conditions, $y(0)=2$ and $y(1)=-3$ is $\frac{d^2y}{dx²}=3x-2$
- A. $y=\frac{x³}{3}-\frac{x²}{2}+3x-6$
- B. $y=3x^3-\frac{x²}{2}-5x+2$
- C. $y=\frac{x³}{2}-x²-\frac{5x}{2}+2$
- D. $y=y³-\frac{x²}{2}+5x+\frac{3}{2}$
Explanation: Integrating the given differential equation $\frac{d^2y}{dx²}=3x-2$ gives $\frac{dy}{dx}=\frac{3x^2}{2}-2x+c_1$. Integrating a second time gives $y=\frac{x³}{2}-x²+c_1x+c_2$. Applying the boundary condition $y(0)=2$ results in $c_2=2$. Applying $y(1)=-3$ gives $-3=\frac{1}{2}-1+c_1+2$, which simplifies to $c_1=-4.5$. The final differential equation solution is $y=\frac{x³}{2}-x²-4.5x+2$. Note that none of the provided options exactly matches this solution.