Question:
What does the differential equation \((2x+y+1)dx+(x+2y+1)dy=0\) represent?
What does the differential equation \((2x+y+1)dx+(x+2y+1)dy=0\) represent?
Show Hint
Check exactness of the ODE, integrate to get an implicit curve, then classify the resulting conic using its discriminant.
Updated On: Jul 3, 2026
- A family of circles
- A family of parabolas
- A family of hyperbolas
- A family of ellipses
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Identify \(M(x,y)=2x+y+1\) and \(N(x,y)=x+2y+1\). Since \(M_y=1\) and \(N_x=1\), the equation is exact.
Step 2: Find \(F(x,y)\) with \(F_x=M\). Integrating \(M\) with respect to \(x\) gives \(F=x^2+xy+x+g(y)\).
Step 3: Differentiate with respect to \(y\): \(F_y=x+g'(y)\). Setting this equal to \(N=x+2y+1\) gives \(g'(y)=2y+1\), so \(g(y)=y^2+y\).
Step 4: The implicit general solution is \[x^2+xy+y^2+x+y=C.\]
Step 5: This is a conic \(Ax^2+Bxy+Cy^2+Dx+Ey+F=0\) with \(A=1,\ B=1,\ C=1\). The discriminant is \(B^2-4AC=1-4=-3<0\), which signals a real ellipse (not a circle, since \(B\neq 0\)).
\[\boxed{\text{Family of ellipses}}\]
Was this answer helpful?
0
0
Top CPET Mathematics Questions
- The value of \(\displaystyle\int_0^{\pi/2} \sin^2(x)\cos^4(x)\,dx\) is ____.
- Which one of the following statements is incorrect?
- What is the surface area of the solid obtained by revolving the curve \(y=\sqrt{9-x^2}\), \(-2\le x\le 2\), about the x-axis?
- Which one of the following equations is obtained from the equation \(xy=-8\) by rotating the axes counterclockwise through an angle of \(45^\circ\)? (Here \(X, Y\) denote the new coordinates.)
- If \(\vec{a}, \vec{b}, \vec{c}\) are three vectors satisfying \(|\vec{a}|=1, |\vec{b}|=2, |\vec{c}|=1\) and \(\vec{a}\times(\vec{a}\times\vec{b}) + \vec{c} = 0\), then what is the angle between the vectors \(\vec{a}\) and \(\vec{b}\)?
View More Questions
Top CPET Ordinary Differential Equations Questions
- If the non-homogeneous term in an ordinary differential equation (ODE) is \(xe^{3x}\) and 3 is a root of the characteristic equation with multiplicity 2, then what is the form of the particular solution (i.e. \(y_p\))? (c is a constant.)
- The Bernoulli equation \(\frac{dy}{dx}+P(x)y=Q(x)y^3\) is best solved by making which of the following substitutions?
- The general solution of the ODE \((D^2+6D+9)y=\frac{e^{-3x}}{x^3}\), \(D=\frac{d}{dx}\) (\(c_1,c_2\) are arbitrary constants) is ____.
- Which one of the following options is true for the initial value problem: \(\frac{dy}{dx}=2y^{1/3}\), \(y(0)=0\)?
- If \(u=x^3\) and \(v=y^2\) transforms the ODE: \(3x^5dx - y(y^2-x^3)dy=0\) into \(\frac{dy}{du}=\frac{\lambda u}{(u-v)}\), then what is the value of \(\lambda\)?
View More Questions
Top CPET Questions
- Which of the following is a key reason for choosing DRAM over SRAM for main memory?
- Which of the following components in an FPGA is responsible for implementing combinational and sequential logic?
- What is the main purpose of declaring a base class as virtual in multiple inheritance?
- What happens if a class contains at least one pure virtual function?
- What happens if a derived class defines a function with the same name as a base class function, but with a different parameter list?
View More Questions