MHT CET · Maths · Differential Equations
The particular solution of the differential equation \(y\left(\frac{\mathrm{d} x}{\mathrm{~d} y}\right)=x \log x\) at \(x=\mathrm{e}\) and \(\mathrm{y}=1\) is
- A \(\mathrm{e}^{x y}=2\)
- B \(x=\mathrm{e}^{\mathrm{y}}\)
- C \(x\mathrm{y}=2\)
- D \(\log x=2 y\)
Answer & Solution
Correct Answer
(B) \(x=\mathrm{e}^{\mathrm{y}}\)
Step-by-step Solution
Detailed explanation
(D)
\(y\left(\frac{d x}{d y}\right)=x \cdot \log x\)
\(\therefore \int \frac{1}{x \cdot \log x} d x=\int \frac{1}{y} d y\)
\(\therefore \log |\log x|=\log y+\log c\)
We have \(x=e\) and \(y=1\)
\(\therefore \log |\log e|=\log 1+\log c \Rightarrow \log c=0\)
\(\therefore \log |\log x|=\log y \Rightarrow \log x=y \Rightarrow x=e^{y}\)
\(y\left(\frac{d x}{d y}\right)=x \cdot \log x\)
\(\therefore \int \frac{1}{x \cdot \log x} d x=\int \frac{1}{y} d y\)
\(\therefore \log |\log x|=\log y+\log c\)
We have \(x=e\) and \(y=1\)
\(\therefore \log |\log e|=\log 1+\log c \Rightarrow \log c=0\)
\(\therefore \log |\log x|=\log y \Rightarrow \log x=y \Rightarrow x=e^{y}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Let
Statement 1 : If a quadrilateral is a square, then all of its sides are equal.
Statement 2: All the sides of a quadrilateral are equal, then it is a square.MHT CET 2023 Easy - If sum of the slopes of lines represented by \(x^2-2 x y \tan \theta-y^2=0\) is 4 , then \(\theta=\)MHT CET 2022 Easy
- In , if tan A + tan B + tan C = 6 and tan A.tan B = 2 then tan C = ……MHT CET 2019 Easy
- The area of the region bounded by the \(y\)-axis, \(y=\cos x, y=\sin x\), when \(0 \leq x \leq \frac{\pi}{4}\), isMHT CET 2022 Medium
- \(\operatorname{Cos}^2 48^{\circ}-\sin ^2 12^{\circ}=\)
,if \(\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}\)MHT CET 2022 Easy - The equation of the tangent to the parabola \(y^2=8 x\), which is parallel to the line \(4 x-y+3=0\) isMHT CET 2024 Medium
More PYQs from MHT CET
- An electric lamp connected in series with a capacitor and an a.c. source is glowing with certain brightness. On increasing the value of capacitance the brightness of the lampMHT CET 2024 Easy
- Choose the correct relation between polarisation 'P' and electric susceptibility \(^{4} \chi_{e}{ }^{\prime}\) of dielectric material. \((\mathrm{E}=\) electric field \()\)MHT CET 2020 Easy
- A gas is compressed at a constant pressure of \(50 \mathrm{~N} / \mathrm{m}^2\) from a volume of \(10 \mathrm{~m}^3\) to a volume of \(4 \mathrm{~m}^3\). Energy of \(100 \mathrm{~J}\) is then added to the gas by heating. Its internal energy isMHT CET 2023 Medium
- The value of \(\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{8}\right)\) isMHT CET 2022 Easy
- The end product in Hofmann's exhaustive of amines isMHT CET 2022 Medium
- If the lines \(\frac{3-x}{2}=\frac{5 y-2}{3 \lambda+1}=5-z\) and \(\frac{x+2}{-1}=\frac{1-3 y}{7}=\frac{4-z}{2 \mu}\) are at right angles, then \(7 \lambda-10 \mu=\)MHT CET 2025 Medium