Nonlinear first order differential equation general solution \frac{d}{dx}y(x)+y(x)^2=ax^2+bx+c

Ellie Castro

Ellie Castro

Answered question

2022-04-24

Nonlinear first order differential equation general solution
ddxy(x)+y(x)2=ax2+bx+c

Answer & Explanation

Giovanny Howe

Giovanny Howe

Beginner2022-04-25Added 18 answers

Step 1
A way to write the general solution to the given first order inhomogeneous ordinary differential equation of Riccati type is in terms of parabolic cylinder functions, Dν(x) is rational with numerator a sum of four weighted D functions and denominator a sum of two weighted D functions:
y(x)=
-4a3/4CDb2+4a3/2-4ac8a3/2(b+2ax2a3/4)-2(2ax+b)D-b2-4a3/2+4ac8a3/2(i(b+2ax)2a3/4)+2C(2ax+b)Db2-4a(c+a)8a3/2(b+2ax2a3/4)-4ia3/4D4a(c+a)-b28a3/2(i(b+2ax)2a3/4)22a(CDb2-4a(c+a)8a3/2(b+2ax2a3/4)+D-b2-4a3/2+4ac8a3/2(i(b+2ax)2a3/4))
Some of the complications in this expression are carefully balanced cancellations that occur when the roots of the right-hand side of the original equation are variously positive, zero, negative, or complex. If you have constraints on those coefficients, the complications might be reduced.

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