Question Type 2: Integrating polynomials with boundary conditions to solve for C Exercises

Given $\frac{dy}{dx}=3x^2$ and $y(0)=5$, find the particular solution $y(x)$.

Determine $y(x)$ if $\frac{dy}{dx}=-3x^4+x^2-2$ and $y(0)=4$.

Given $\frac{dy}{dx}=5x^4$ and $y(1)=3$, find $y(x)$.

Given $\frac{dy}{dx}=2x^3-3x^2+4x-5$ with $y(2)=1$, find the constant $C$ and the expression for $y(x)$.

If $\frac{dy}{dx}=4x^3+2x$ and $y(0)=2$, determine $y(x)$.

Find the particular solution $y(x)$ given $\frac{dy}{dx}=8x^4+2x^3+11x+3$ and the condition $y(1)=9$.

If $\frac{dy}{dx}=5x^4-x^3+2x^2-x+1$ and $y(0)=-1$, determine $y(x)$.

Given $\frac{dy}{dx}=3x^2+4x+6$ and $y(-2)=7$, find $y(x)$.

If $\frac{dy}{dx}=x^5-2x^3+x$ and $y(-1)=0$, find $y(x)$. [5 marks]

Find $y(x)$ if $\frac{dy}{dx}=6x^2-4x+1$ and $y(1)=5$.

Find $y(x)$ given $\frac{dy}{dx}=2x^6-x^4+3x^2-5$ and $y(1)=0$.

Find the particular solution for $\frac{dy}{dx}=7x^3+9x^2-x+8$ with $y(1)=10$.