Textbook is Differential equations for scientists and engineers 2017 by Constanda.

C. Constanda (2017) textbook, Section 2.3, ## 3, 7

2.3. Solve the given IVP or find the GS of the DE if no IC is given.

3. 3xy'= x + 3y.

7. (x^2)y' 2xy - y^2 = 0, y(1) = 1.

C. Constanda (2017) textbook, Section 2.6, #5

Solve the given IVP or find the GS of the DE if no IC is given.

5. 2 + 3y + (ye)^x/2 + (3x 2y + 2e^x/2)y= 0, y(0) = 1.

The answers to question 3, 7 and 5 is:

• y(x) = x(ln|x| + C)/3
• y(x) = x^2/(2-x)
• 2x + 3xy y2 + 2y^ex/2 + 3 = 0.

*** Please explain how these answers were derived.****

2. Find the largest open interval on which the conditions of Theorem2.29are satisfied, without solving the IVP itself.

• (t^2 3t + 2) y'+ ty = e^t, y (3/2) = 1.

The answers to question 2 is:

• 1 < t < 2

*** Please explain how this answer was derived.****

3. Indicate the regions in the (t, y)-plane where the conditions of Theorem 2.37 are not satisfied.

• (2y + t)y' = 2t + y.

The answers to question 3 is:

• {(t, y) : y = t/2, < t < }.

*** Please explain how this answer was derived.****

C. Constanda (2017) textbook, Section 2.7, ## 3, 7, 13

4. Solve the DE with each of the given ICs and find the maximum interval of existence of the solution.

• y = 4ty^2; y(0) = 2; y(1) = 2; y(3) = 1.

The answers to question 4 is:

• y(t) = 2/(1 4t2), 1/2 < t < 1/2; y(t) = 2/(3 4t2), < t < 3/2;
• y(t) = 1/(17 2t^2), 17/2 < t < .

*** Please explain how this answer was derived.****

5. Sketch the direction field for the equation y'(x)=x^2. Find the solution corresponding to the initial condition y(0)=1 and plot this solution curve within the direction field plot.

6. Sketch the direction field for the equation y'(x)=-2*y(x). Find the solution corresponding to the initial condition y(0)=-3 and plot this solution curve within the direction field plot.