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Q1:Write the second order inhomogeneous linear differential equationSee Answer
Q2:16. sin(xy) = cos(x + y) 17. √√x+y=x² + y²¹ Answer + 18. sin x cos y = x² — 5y 19. √√xy=1+x²y Answer 20. xy = √x² + y²See Answer
Q3: 2) Solve the given third-order differential Non-Homogeneous equation by variation of parameters: y^{\prime \prime}+5 y^{\prime}+3 y=0See Answer
Q4: 4) Solve the ODE by using Superposition Method: y^{\prime \prime}+3 y=x+3 e^{-3 x}See Answer
Q5: 5) Solve the Non-linear second ODE: 3 x^{2} y \frac{d^{2} y}{d x^{2}}+y^{2}=x^{2}\left(\frac{d y}{d x}\right)^{2}See Answer
Q6: Determine the differentials with respect to x of the following. Take care to show your method clearly and to simplify the results.1. \text { a) } \sin (x) \cdot \cos (x) \text { b) } \ln \left(3 x^{2}\right) \text { c) } x \sqrt{2 x^{2}+14} \text { d) } \frac{e^{x}-e^{-x}}{2} \text { e) } \tan (x) \text { f) } 6 a x^{3}-4 a cSee Answer
Q7: Find the value of x at the turning points on the curve of y=x^{3}-6 x^{2}+9 x-2See Answer
Q8: • Perform the indicated operation. Write the answer in lowest terms. \frac{a^{2}-5 a+4}{a^{2}-a-12} \div \frac{a-4}{a^{2}+5 a+6}See Answer
Q9: \text { 9. } Y(s)=\frac{2}{s-5}-\frac{3 s}{s^{2}+4}+\frac{1}{s^{2}}See Answer
Q10: \text { 14. } \quad y^{\prime}+2 y=4 \sin (3 t), \quad y(0)=3See Answer
Q11: \text { 3. } y^{\prime \prime}+4 y^{\prime}+8 y=\cos (4 t), \quad y(0)=0, \quad y^{\prime}(0)=0See Answer
Q12: \text { 10. } Y(s)=\frac{2}{s+5}-\frac{3}{s^{2}+9}-\frac{10}{s^{5}}See Answer
Q13: \text { 13. } y^{\prime \prime}+4 y=\cos (t), \quad y(0)=2, \quad y^{\prime}(0)=0See Answer
Q14: \text { 1. } y^{\prime \prime}+2 y^{\prime}+2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=0See Answer
Q15: \text { 11. } Y(s)=\frac{2}{2 s+5}-\frac{3 s}{s^{2}+2}+\frac{1}{5 s^{2}}See Answer
Q16: \text { 12. } Y(s)=\frac{2}{3 s-5}-\frac{3}{2 s^{2}+4}+\frac{1}{3 s}See Answer
Q17: Use Laplace transforms to solve each of the differential equations 5 - 8, in terms of the initial conditions y(0) and y'(0). Compare your solution to.the general solution you would obtain using the methods of Chapter 3 (identify the constants C1 and C2). \text { 6. } y^{\prime \prime}+2 y^{\prime}+5 y=0See Answer
Q18: Calculate the maximum volume of an open-toped box that can be made from a sheet of cardboard 47 cm long by 38 cm wide. See Answer
Q19: \text { Let } a<b \text { and let } f:[a, b] \rightarrow[0, \infty) \text { be a continuous function. } Decide whether or not the following statements are true or false. Please explain your answer! \text { a) If } f \text { is Lipschitz, then } f^{2} \text { is Lipschitz. } \text { (b) If } f^{2} \text { is Lipschitz, then } f \text { is Lipschitz. }See Answer
Q20: \text { Let } f:[0, \pi] \rightarrow \mathbf{R} \text { be given by } f(x)=\left\{\begin{array}{ll} -2-\sin (2 x) & \text { for } x \in\left[0, \frac{\pi}{2}\right) \\ 2 x-\pi & \text { for } x \in\left[\frac{\pi}{2}, \pi\right) \\ 2 & \text { for } x=\pi \end{array}\right. \text { (a) For } x \in[-\pi, \pi] \text {, draw the graph of the Fourier sine series of } f(x) \text {. } \text { (b) For } x \in[-\pi, \pi] \text {, draw the graph of the Fourier cosine series of } f(x) \text {. } (c) Please explain, in words, how you know that your answer in (b) is indeed the graph of the Fourier cosine series.Fourier coefficientsSee Answer

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