2.1E: Integrals as Solutions (Exercises)

In Exercises 1-4, solve the following initial value problems.

1. \(\frac{dy}{dx}=x^2+x \quad \text{with initial condition}\quad y(1)=3\).

2. \(\frac{dy}{dx}=\sin(5x)\quad \text{with initial condition}\quad y(0)=2\).

3. \(\frac{dy}{dx}=\dfrac{1}{x^2-1} \quad \text{with initial condition}\quad y(0)=0\).

4. \(\frac{dy}{dx}=e^x+x \quad \text{with initial condition}\quad y(0)=10\).

In Exercises 5-12, solve the following differential equations.

5. \(\frac{dy}{dx}=xe^x \).

6. \(\frac{dy}{dt}=t\sin(t^2) \).

7. \(\frac{dy}{dt}=\dfrac{t}{t^2+4} \).

8. \(\frac{dy}{dx}=\dfrac{1}{x^2+4} \). 

9.  \(y''=x+4 \).

10. \(y''=e^{2x}+x \).

11. \(y'''=2x+e^x \).

12. \(y'''=\sin(x) \).

 

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