Y'+y^2=25 solve for general solutionFind the order of the differential equationFind whether the differential equation is linear or not linear

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Y&#039;+y^2=25 solve for general solutionFind the order of the differential equationFind whether the differential equation is linear or not linear

nick1337 · Accepted Answer

To solve the given differential equation, let&#039;s denote the dependent variable as y and the independent variable as x. The differential equation is:y′+y2=25To find the general solution, we need to solve this differential equation.First, let&#039;s rearrange the equation:y′=25−y2This is a first-order nonlinear ordinary differential equation. To solve it, we can use separation of variables.Separating the variables, we have:dydx=25−y2Now, we can rewrite the equation in a more convenient form:dy25−y2=dxNext, we integrate both sides of the equation. On the left side, we need to use partial fraction decomposition to simplify the integration. The decomposition can be written as:125−y2=A5+y+B5−yMultiplying through by 25−y2, we have:1=A(5−y)+B(5+y)Expanding and collecting like terms, we get:1=(A+B)+(B−A)yEquating the coefficients of the constant and y terms, we find:A+B=1  (Coefficient of the constant term)B−A=0  (Coefficient of the y term)From the second equation, we have B=A. Substituting this into the first equation, we get 2A=1, which implies A=12 and B=12.Therefore, the partial fraction decomposition becomes:125−y2=12(5+y)+12(5−y)Now, we can integrate both sides of the equation:∫125−y2dy=∫(12(5+y)+12(5−y))dxThe left side can be integrated using partial fractions. The right side can be integrated using simple substitution.The integral on the left side gives:12ln|5+y5−y|=12ln|y+55−y|On the right side, we have:12ln|5+y|−12ln|5−y|=12ln|5+y5−y|Combining both sides of the equation, we get:12ln|y+55−y|=12ln|5+y5−y|Now, we can drop the absolute value signs, as the expression inside the logarithm is always positive.ln(y+55−y)=ln(5+y5−y)The natural logarithm of the same quantity is equal, so we can equate the arguments inside the logarithms:y+55−y=5+y5−yCross-multiplying, we obtain:(y+5)(5−y)=(5+y)(5−y)Expanding and simplifying, we have:25−y2=25−y2This equation is always true, indicating that the solution is valid for all y.Therefore, the general solution to the given differential equation is:ln(y+55−y)=Cwhere C is the constant of integration.

Y'+y^2=25 solve for general solutionFind the order of

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