Solve the following ordinary differential equation by the method of separation o

Question

Solve the following ordinary differential equation by the method of separation of variables. dz/dt = x(2 - y2)3/y when y(0) = 1 Recall that some differential equations can be written in the form q(y) y' = p(x) When this happens then one can write q(y) dy = p(x)dx + C and have an implicit solution. In some cases this the implicit solution can be arranged to find an explicit solution y(x). What is q(y) in this case? q(y) = Enter just an expression for q(y). Do not include y' What is p(x) in this case? p(x) = Enter just an expression for p(x). Calculate an antiderivative of q(y)? q(y) dy = Enter just an expression in y. Do not include a constant of integration! Calculate an antiderivative of p(x)? p(x) dx = Enter just an expression in x. Do not include a constant of integration! What is the value of the constant C in the equation q(y) dy = p(x) dx + C? C = Calculate an exact number C using the given information, e.g. ln(8)/3. Do not give a decimal answer. What is the explicit solution of the given differential equation? Enter an equation in x and y, but do not solve for y.

Explanation / Answer

1.q(y)= y/(2-y^2)^3

2. p(x)=x

3. take y squared = z so the derivative becomes q(z)= dz/(2-z)^3 which, when integrated becomes 2/(2-z)^2 . replacing z by y square we get the anti derivative = 2/(2-y^2 )^2

4. antiderivative of p(x) is simply x^2/2.

5. the expression becomes, 2/(2-y^2)^2 = x^2/2 + C. Now given that y(0) = 1. putting x=0 and y= 1 in the equation we get, C=2.

6. the implicit solution is therefore, 2/(2-y^2)^2 = x^2/2 +2

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