Question:
The solution of the differential equation \( \frac{dy}{dx} = \sin(x+y)\tan(x+y) - 1 \) is
The solution of the differential equation \( \frac{dy}{dx} = \sin(x+y)\tan(x+y) - 1 \) is
Show Hint
For equations with $(x+y)$, substitute $v = x+y$ to transform it into a separable equation.
Updated On: Apr 10, 2026
- $\csc(x+y) + \tan(x+y) = x + c$
- $x + \csc(x+y) = c$
- $x + \tan(x+y) = c$
- $x + \sec(x+y) = c$
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Substitution
Put $x+y = z \implies 1 + \frac{dy}{dx} = \frac{dz}{dx}$. Equation becomes $\frac{dz}{dx} - 1 = \sin z \tan z - 1 \implies \frac{dz}{dx} = \sin z \tan z$.
Step 2: Variable Separation
$\frac{dz}{\sin z \tan z} = dx \implies \frac{\cos z}{\sin^{2}z} dz = dx$.
Step 3: Integration
$\int \csc z \cot z dz = \int dx$. $-\csc z = x - c \implies x + \csc(x+y) = c$.
Final Answer: (b)
Put $x+y = z \implies 1 + \frac{dy}{dx} = \frac{dz}{dx}$. Equation becomes $\frac{dz}{dx} - 1 = \sin z \tan z - 1 \implies \frac{dz}{dx} = \sin z \tan z$.
Step 2: Variable Separation
$\frac{dz}{\sin z \tan z} = dx \implies \frac{\cos z}{\sin^{2}z} dz = dx$.
Step 3: Integration
$\int \csc z \cot z dz = \int dx$. $-\csc z = x - c \implies x + \csc(x+y) = c$.
Final Answer: (b)
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Differential equations Questions
- The general solution of $x\sqrt{1+y^{2}}\,dx + y\sqrt{1+x^{2}}\,dy = 0$ is
- By eliminating the arbitrary constants $A$ and $B$ from $y = Ax^{2} + Bx$, the differential equation obtained is
- If $x(1+y^{2})\,dx + y(1+x^{2})\,dy = 0$ and $y(0) = 1$, then $x^{2}y^{2} + x^{2} + y^{2}$ equals
- The differential equation of the family of curves $y = a\cos\mu x + b\sin\mu x$, where $a$ and $b$ are arbitrary constants, is
- The equation of the curve passing through the origin and satisfying $\dfrac{dy}{dx} = (x - y)^{2}$ is
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions