problem 1.4-2: ============= There are two answers in the answer report. Show the steps for the equil sol y=0 and the non-equil sol y=???. An answer check is required. The easiest answer check is for the explicit sol. problem 1.4-6: ============= The problem splits into two problems (1) y' = 3sqrt(x)sqrt(y) for x>0, y>0, and (2) y' = 3sqrt(-x)sqrt(-y) for x <0, y<0 There is one equil sol y=0. Show the sol steps for the equil sol, and for each of (1),(2). That makes 3 panels of solution, which makes for 3 answers in the answer report. Check the answers to (1), (2). The equil sol y=0 has reversible steps (document your ans check accordingly). Ans equilibrium sol: y=0 Ans quadrant I: B.o.B. ans y = (x^(3/2)+c1)^2 Ans quadrant III: y = -((-x)^(3/2)+c2)^2 problem 1.4-8: ============= The separable form is cos(y)y'=2x Equil solutions arise from 2x sec(y)=0 which has no solution y=K=constant. The answer report has one answer, which is implicit. An answer check for the implicit solution is expected. An explicit solution y=arcsin(x^2+c) has been found by many people, but it is not equivalent to the implicit solution: the answer check only works with restrictions on x,c. To understand the difficulty, consider the equation sin(u+4Pi)=0.67. Do you get the answer u+4Pi=Arcsin(0.67)? Does not this equation imply that u+4Pi is between -Pi/2 and Pi/2? problem 1.4-10: ============= The book's answer is wrong, it is missing a term of -1. The correct answer is two equations, y = -1 and y = -1 + (x+1)/(1+(x+1)c). An answer check is required for the second solution above. The equilibrium solution y=-1 answer check consists of the phrase "reversible steps." problem 1.4-18: ============= Let y'=f(x,y), f(x,y)=(1-x^2+y^2-x^2y^2)/x^2. Then f(2,0)=-3/4 and F(x)=f(x,0)/f(2,0), G(y)=f(2,y) converts y'=f(x,y) into y'=F(x)G(y), the separable form. Divide by G(y) to get the separated form y'/G(y)=F(x), then apply quadrature. Equil solutions y=K arise from G(K)=0. Solve this equation for K and report all equil sols. Be careful about the answer check, if the steps are not obviously reversible. The implicit answer involves arctan(y). Answer check the implicit non-equil solution, if you wish (see below). Is there an explicit solution? Yes, and you can find it by applying the tangent to each side of the implicit equation. It works, because the tan(arctan(u))=u and sec^2(u)=1+tan^2(u). Please report the explicit solution and do an answer check. problem 1.4-22: ============= y'=4x^3y-y, y(1)=-3 There are two equations (DE+IC) and hence two answer checks. The equil sol is y=0 and since it cannot satisfy y(1)=-3, then only the non-equil solution needs to be derived and checked. The answer that results has a constant c in it, which is determined by y(1)=-3. Show all steps in obtaining the non-equil sol and do an answer check. Then find c and check that the proposed solution satisfies y(1)=-3. Is there an explicit solution? Yes, because ln|y|=expression in x. Use properties of log and exp to find the explicit solution. That formula should be reported with c evaluated, followed by an answer check of the DE and the IC.