7.4 Workshop Exercises 4

For more worked examples, see Gilbert and Jordan Guide to Mathematical Methods, pages 153–164.

The main themes this week are: double integration (W4.1-4.2); 2nd order homogeneous differential equations, constant coefficients (W4.3-4.4), integrable differential equations (W4.5-4.6); separable differential equations (W4.7-4.8); linear first-order differential equations (W4.9-4.10). For questions W4.3-4.4, see slides 4.45-49 in the MATH101 notes.

W4.1. Calculate the repeated integrals

and

In each case, start by sketching the region of integration.

W4.2. Evaluate the double integral $\int_{0}^{\frac{\pi}{6}}\int_{0}^{\frac{\pi}{6}}\sec^{2}(x+y)\,dx\,dy$.

W4.3. Find the general solutions to the following differential equations:

a) $\frac{d^{2}y}{dx^{2}}+4\frac{dy}{dx}+3y=0$.

b) $\frac{d^{2}y}{dx^{2}}+4\frac{dy}{dx}+4y=0$.

c) $\frac{d^{2}y}{dx^{2}}+4\frac{dy}{dx}+5y=0$.

W4.4. Solve the following initial value problems:

a) $\frac{d^{2}y}{dx^{2}}-3\frac{dy}{dx}-4y=0$, $y(0)=1$, $y^{\prime}(0)=9$.

b) $\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}+10y=0$, $y(0)=0$, $y^{\prime}(0)=1$.

W4.5.Find the general solution of the differential equation $(x+1)\frac{dy}{dx}=x+2$.

W4.6. Solve the initial-value problem

(*) What is the maximum possible range of values of $x$ on which this defines a valid solution?

W4.7. Find the general solution of the separable differential equation $\frac{dy}{dx}=y(x^{2}+1)$. Find the particular solution with $y(0)=3$.

W4.8. Find the general solution to the differential equation $\frac{dy}{dx}=\frac{x}{3(y+1)^{2}}$.

W4.9. Solve the initial value problem $(x^{3}+1)\frac{dy}{dx}-3x^{2}y=x^{2}$, $y(0)=1$.

W4.10.Put the differential equation

into the standard form of a first-order linear equation, and find the general solution.